Odo has a working communicator:
Is that a part of himself (like his clothes)? Or is it real but he conceals it?
I imagine it is the latter (would be hard for him to create) but I don't know if it is ever addressed.
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It is real. I remember at least one scene in which he took off the communicator and set it down on his desk in the security office because he didn't want anyone to know where he was going.
Although it does still beg the question of what he does with it when he shape shifts. My guess would be that he just carries it inside himself in whatever shape he creates.
In the episode "Broken Link", when Odo is losing control of his shape shifting, there are a couple of shots where you can see his communicator is partially liquefied, indicating it is actually part of him, and not a device.
Odo stated that if he became a rock, all sensors would indicate he is a rock. This means that when he becomes a rock, he is actually a rock, and not just a facsimile of the rock (i.e. the composition is also a rock). Nothing says he can't actually become a complex piece of electronics like a communicator. He can turn into fire. Why not a communicator, with practice?
In episode 3 "Past Prologue", Odo becomes a mouse. So he can replicate more than just "dead" matter.
I just started watching the series on Netflix and happened to catch it when he was observing the Klingon women. Most likely the communicator is held within the shape he has assumed at the time.
I have no reference but AFAIR, Odo can only replicate "dead" matter (like a container) or "simple" machines (maybe a piston or a sliding door) - not complex electronics.
He can look like a wall monitor (and probably even shift the colors on his surface and respond to touch, especially when they look as simple as the command interfaces ones used in Star Trek) but he can't send commands over a wire or receive a TV stream or the like.