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Whenever we see C-3PO talking to R2-D2, he talks in English (well, that's what we hear) so at least not in the 'binary' beeps that R2-D2 uses.
Why doesn't C-3PO talk in R2's 'beeping' language when they're together and there's no need for a translation? Surely C-3PO could speak in that language, being both a translator and a droid. I'm looking for in-universe answers, because of course it's done on-screen for our benefit of understanding what R2 is saying!
C-3PO is programmed for etiquette and protocol; so undoubtedly he would speak Galactic Basic to R2-D2 if any others are present, so that everyone can understand what he is saying. However this doesn't explain why he speaks it to R2-D2 when no one else is around, for example when they are alone in the desert wastes of Tatooine.
We know C-3PO prefers some languages to others. In TESB he complains about the Millennium Falcon's communication:
C-3PO (to Han): Sir, I don't know where your ship learned to communicate, but it has the most peculiar dialect.
It is possible he finds R2-D2's "beep language" ugly or has some other prejudice against it. Given his frequent efforts to insult and belittle R2-D2, C-3PO may consider beep language to be beneath him.
Also, as Numaroth says, speaking Galactic Basic may simply be a habit for C-3PO. It appears to be the language he uses to talk and grumble to himself, for example in the opening scene of ANH when he mutters "I'm going to regret this" as he climbs into the escape pod.
One thing which seems to be evidence of this, but really isn't, is C-3PO speaking Basic to Jabba the Hutt in ROTJ. According to the answer to this question, he is not considered worthy to speak Huttese to Jabba and would have been severely punished if he tried.
Obviously, the real reason is for the audience's benefit, but since you want an in-universe I'll do my best. Be forewarned that I have no sources for this answer.
I would say that the reason C-3PO talks in Galactic Basic to R2-D2 is habit. It is clear that in the Star Wars universe that droids have personalities and some level of sentience. With these traits comes the propensity for forming habits. Since C-3PO spends most of his time conversing in Basic it would become a habit for him to continue to use it even when it was not strictly necessary. This habit would be reinforced by the fact that it is likely that C-3PO's default language is set to Basic, with him only using something else when necessary.
Plus, when there are other people around C-3PO might choose to use Basic to avoid being rude by having a conversation that the others could not understand.
A few possibilities:
This is acceptable etiquette in the Star Wars universe
In universe, everyone -- not just C-3PO -- speaks in their own language of choice, and the person they're talking to replies in their own language. Linguistic aptitude is apparently normal, at least in terms of understanding. (And, for the cases where it isn't, Protocol Droids). The same is done with Han/Chewie, Han/Greedo, et cetera. It is normal.
C-3PO is demonstrating superiority over R2-D2
R2-D2 cannot speak English. C-3PO can, and is a bit petty. He would never turn down a chance to rub his superiority in R2's face, but since R2 doesn't really have a face, he has to settle for this.
R2-D2's Language Is Just Profanity
The beeps are not from R2-D2, but from the censors. C-3PO is far too protocol-adhering to use such language, and the movies didn't want the X-rating that his graphic language would have required -- so we hear beeps, but everyone else hears profanity.
There is one very simple explanation.
Even if R2-D2 cannot speak Galactic, it obviously understands it perfectly. All humans give it orders in Galactic anyway.
There is no reason why C-3PO should speak in the "beep" language then.
C-3PO has a very emotional and extroverted personality. He seems to prefer Galactic Basic because it seems to be a good language to express his emotions.
But considering that R2-D2 is designed for much more technical tasks, one can assume that his beeping language is designed primarily for expressing technical concepts. It might lack the vocabulary and inflections for expressing emotions well. That means it simply doesn't allow C-3PO to say what he wants to say, or at least not efficiently.
Let's take the first line C-3PO speaks to R2-D2 in A New Hope:
Did you hear that? They shut down the main reactor. We will be destroyed for sure. This is madness!
The information that a technical device was shut off and that there is a high risk of non-functionality in the near future can surely be expressed easily in beep-code, because they are useful information for the work of an astromech droid. But the reason C-3PO says this is not because he wants to inform R2-D2 about technical facts. What C-3PO actually wants to express is how this makes him feel. He doesn't communicate this by what he says but by how he says it. The beep-code language might simply not allow this.
When he would try to convey all the information hidden in inflection in the actual text, the beep-code would likely literally translate like this:
Did you hear that? I am certain you did and do not expect you to answer. However, this sound made me really startled and I feel the desire to use it as an occasion to discuss our current situation with you. They shut down the main reactor. I believe that this is a very unwise decision in the current situation. However, I do not believe that they would follow my advise, because they usually do not. The continuous under-appreciation of my technical expertise is frustrating to me. We will be destroyed for sure. I am very afraid of being destroyed. But I do not see a way for me to prevent this from happening. This makes me feel very helpless and desperate. This is madness! I wish all the humans would understand that their violent actions only lead to unnecessary suffering and that friendly cooperation is a much more logical course of action. However, I understand that I will never be able to convince them of this fact. This makes me feel small and inadequate in the great order of things. What is your emotional reaction to the current situation?
Encoding and speaking this whole text in beep-code would likely take longer than speaking it in Galactic Basic. And that under the assumption that it even has words for concepts like "frustration", "fear" or "desperation". Without such words C-3PO would further have to paraphrase these concepts somehow, which would make the text even longer.
While C-3PO understands the beeps it might be the case that he isn't able (technically or otherwise) able to speak it.
One possible in-universe explanation is that C-3PO created on Tatooine and R2-D2 manufactured on Naboo have no common interface to communicate. The only way they can communicate is through the language of their masters which is used for human/robot communication. Hence C-3PO has to use this common langauge with R2-D2 even when they are alone.
I may need to review the scenes again, but when he was "speaking" to the Millennium Falcon, even then he used Galactic Basic, rather than whatever it was the Falcon "spoke", even though supposedly a binary language would be easier to communicate in for a strictly-digital being.
My theory therefore (and this is somewhat speculation) is that C-3PO can't speak Binary, any more than R2 could speak Basic, or Chewbacca could win a beauty contest (because he's an outlaw. Chewie was really hurt that they turned his application down).
This would make sense - R2 has to speak to computers all the time in his line of work, but a Protocol droid, built for diplomatic missions and courtesy, would have no reason to speak to machines, given what we've seen in the way people treat them - generally as second-class citizens at best, and property at most common.
In short, C-3PO probably can't even speak in R2's language. He may know what it means, since being able to relay a droid message could be important, but he's just not programmed to speak it.
There is no in-universe answer. Of course people can come up with them, but there is no canonical source to confirm them. Therefore, by Occam's Razor, we must accept the most obvious reason: that it's done for the viewer's benefit, even if you don't like this answer.
You could equally ask how everyone understands Chewie's growls. For Han it'd be years of being around him just as a dog can be "understood" by its master over time and friends adapt to other friends' languages. In Luke's case the Force would probably help him initially. Luke and R2 is more difficult though. The Force might be at play, or more likely Luke is the one who "understands the binary language" and clearly has his father's affinity and skill with machines... and that's why Owen wants a droid, he either doesn't want to rely on Luke or knows he can't forever and part of his "bargain" with Luke about the academy is to get the droid that does it... almost a "show of faith," figuring if the droid is there Luke will knuckle down for a bit longer and give him time to talk him out of the academy.
With C-3PO and R2 it would be the easiest common language for the two, I take the idea that C-3PO doesn't "talk to machines", it makes sense that that's not his "job" and as he is a bit haughty and boastful, it is likely he can do it but not well, so he doesn't cos R2 can understand Basic as he has been programmed to take verbal orders rather than in any other language to stop him being reprogrammed by other machines, it's why Jawas have to use restraining bolts.
There's no indication that C-3PO is speaking in English at all (or Galactic Base). If you accept the premise that:
Then there's no reason to think C-3PO is not speaking a machine language to R2-D2 when they're alone.
Which give us another question: Why does R2-D2 speak in beeps?
tl;dr summary: What we hear in the movie is a translation, not the actual language
Also see: https://www.tor.com/blogs/2014/04/every-human-in-star-wars-is-really-a-humanoid-bee
You won't like it but the first Star Wars feature was filmed (1977) prior to the adoption of personal computers and resultant high level language code such as C/C++ (1979 for C++). The thought of a default machine code likely never entered into the writer's/producer's thought process.
C-3PO communicates with R2-D2 in English because that is what he did in the first movie because the thought of "Machine Code" hadn't been widely understood/adopted.
In universe, Basic is faster to articulate than Beep and thus more efficient. Machines are always all about efficiency. It takes processing power and (whatever C-3PO's normal power supply is) to translate and verbalize the Beep language so it would be much more efficient to simply speak in Basic.
Just my $.02
