As per Show, we can't deduce if the Westerosi peasants know/care about Daenerys and her dragons because show usually has to cover the noble class in their limited time.
But since you are talking only about Westerosi and make no distinction between classes, the answer is a categorical yes. Some Westerosi Nobles do know about Daenerys and her dragons but however they do not yet know for sure about their size right now.
To rephrase it, some Westerosi people know about Daenerys, her ascent and her Dragons but they do not know that Dragons have grown up. (But this would be implied because since they haven't heard that the dragons died, it must be obvious that they grew up).
First, Varys told Tyrion about that in Season 2 Episode 8:
Varys: Daenerys Targaryen lives.
Tyrion: A girl at the edge of the world is the least of our problems.
Varys: She has three dragons.
Tyrion: But even if what they say is true, it'll be years before they
are fully grown.
Varys: And then there will be nowhere to hide.
Tyrion, being a cynical man, somewhat doubted it.
In Season 4 Episode 6, Varys makes his reports about Daenerys and her dragons to the small council.
Varys: More whispers from the east, my lord.
Tywin: The Targaryen girl?
Varys: Daenerys has taken up residence in Meereen. She has conquered
the city and rules as its queen.
Cersei: Conquered with what?
Varys: She commands an army of Unsullied, my queen. Some 8,000
strong. She has a company of sellswords -- the Second Sons. She has
two knights advising her-- Jorah Mormont and Barristan Selmy. And she
has three dragons.
Cersei: Baby dragons.
Varys: Larger every year, Your Grace.
Pycelle: Mormont is spying on her for us?
Varys: No longer. He appears to be fully devoted to her.
Later what Tywin said proves that he believed Varys:
Dragons haven't won a war in 300 years. Armies win them all the time.
She must be dealt with.
But do they know about the extent to which the dragons have grown now? Not that we know of.
The above were the last known reports of Daenerys and her dragons to Westerosi lords and they are pretty much outdated now. Tywin is dead, Tyrion is with Daenerys. Varys has gone over to Dany as well so the spying chain of Iron Throne is still broken, with Qyburn trying to get it running once again. So it is unlikely that Iron Throne received any news about Dany afterwards.
Since Iron Throne did not deal with Dany or her dragons back then, All the Lords & ladies who were aware of that knowledge must have inferred that by now, Dragons must be larger than mere hatchlings that they were at the time of initial reports.
From Books perspective, there are rumors in Westeros regarding the dragons but there are mixed opinions about authenticity of the claims. There is no precise mention of size of dragons in the said rumors. These rumors are mostly in port cities like Oldtown, White Harbor, Kingslanding (Doubtably) & Lannisport (Doubtably) because of the continuous interaction with Sailors coming from Essos, especially Slavers Bay.
For Example from AFFC Prologue, these are students at Citadel talking, some of whom are noble, some are common born:
“How do you know they didn’t [Live in a fish belly]?” Mollander
thumped through the grass, looking for more apples. “You’d need to be
down the belly yourself to swear they weren’t. One sailor with a
story, aye, a man might laugh at that, but when oarsmen off four
different ships tell the same tale in four different tongues...”
“The tales are not the same,” insisted Armen. “Dragons in Asshai,
dragons in Qarth, dragons in Meereen, Dothraki dragons, dragons
freeing slaves... each telling differs from the last.”
“Only in details.” Mollander grew more stubborn when he drank, and
even when sober he was bullheaded. “All speak of dragons, and a
beautiful young queen.”
And they did know about Daenerys:
“No dragon has ever had three heads except on shields and banners,”
Armen the Acolyte said firmly. “That was a heraldic charge, no more.
Furthermore, the Targaryens are all dead.”
“Not all,” said Alleras. “The Beggar King had a sister.”
“I thought her head was smashed against a wall,” said Roone.
“No,” said Alleras. “It was Prince Rhaegar’s young son Aegon whose
head was dashed against the wall by the Lion of Lannister’s brave men.
We speak of Rhaegar’s sister, born on Dragonstone before its fall. The
one they called Daenerys.”
“The Stormborn. I recall her now.” Mollander lifted his tankard high,
sloshing the cider that remained. “Here’s to her!”
And they did know what she had managed to do:
“Truly? Then how would I tell you about the dragons?” Leo shrugged
again. “The mongrel has the right of it. The Mad King’s daughter is
alive, and she’s hatched herself three dragons.”
If Leo Tyrell knows, that means there is a fair chance that his father knows as well and so does Lord Hightower.