Shakespeare's contributions to the English language have been exaggerated. When dictionaries first started being drafted, especially the Oxford English Dictionary, a lot of words were credited to Shakespeare as first known usage. But just because he was the first known usage available to them doesn't mean he was really the first (and, to be fair, there was never a time when the drafters claimed that was the case.) Modern computer analysis of contemporary works are regularly reducing the number of words thought to be coined by Shakespeare. Those studies also contradict the idea that his vocabulary was unusually large. It was a verbose age and Shakespeare's vocabulary seems to be about on par with his contemporaries.

Of course, Shakespeare used the same plots as many of his contemporaries so it's not odd that he used the same words. It's what he did with them that was amazing.

That being said Shakespeare's unusual ability to craft words was noted by contemporaries. He was not held in awe as he would be later but it was certainly noted that he was unusually talented. Ben Jonson gives a criticism of Shakespeare that many high school students would share (that maybe he could have done with fewer words.) But in doing so he admits that the people who knew him were amazed by his ability to put pen to paper and craft these wonderful words without having to edit himself.

I remember the players have often mentioned it as an honor to Shakespeare, that in his writing, whatsoever he penned, he never blotted out a line. My answer hath been, 'Would he had blotted a thousand,' which they thought a malevolent speech. I had not told posterity this but for their ignorance, who chose that circumstance to commend their friend by wherein he most faulted; and to justify mine own candor, for I loved the man, and do honor his memory on this side idolatry as much as any. He was, indeed, honest, and of an open and free nature; had an excellent fancy, brave notions, and gentle expressions, wherein he flowed with that facility that sometime it was necessary he should be stopped. 'Sufflaminandus erat,' as Augustus said of Haterius. His wit was in his own power; would the rule of it had been so too.

All emphasis added. Jonson is responding to what he suggests is common praise of Shakespeare's wit, ability to draft "gentle expressions" and general ability with the language. That certainly suggests that the people Jonson are referring to knew there was something unusually good about Shakespeare. And Jonson, himself, concedes he had exceptional wit but he didn't have to use all of it every time.

He was also the second playwright to be honored with a folio collection. The first was Ben Jonson who sought his folio collection himself. Shakespeare was dead when his was done. Meaning he was the first playwright of the Elizabethan/Jacobean era that his contemporaries decided must be preserved.

In his eulogy to Shakespeare Jonson also notes that "[a]nd though thou hadst small Latin and less Greek" (Jonson can't help but throw in a small burn there) he would rank Shakespeare above the greats of Roman and Greek culture and Britain triumphs because of him. It's a eulogy but Jonson was competitive and jealous as well as reverential of Shakespeare as well as very grounded in the classics. He probably didn't really mean His conceding that Shakespeare's works rank among that greats of Greek and Roman literary achievements is significant. He may be exaggerating slightly as was does in a eulogy but, at the same time, this comparison means a lot to someone like Jonson.

So, his contemporaries wouldn't have had our awe for his language. Because his language wasn't nearly as unique as we have been lead to believe. But that he was doing incredible things with those words was acknowledged on some level. At least among the few in the theater world.