Shakespeare was well known by fellow poets and playwrights during his life, and was given varying degrees of respect by them. He was not regarded anywhere close to the way he is now, that process would not begin until the Enlightenment period and would reach its zenith during the Romantic movement.

We can look at arguably the two most famous of Shakespeare's contemporaries, both during their time and today, to see how Shakespeare was perceived. Christopher Marlowe, widely considered to be one of the greatest playwrights of his time, had a bit of a rivalry with Shakespeare, and there was a significant exchange of ideas between the two*. In "Shakespeare's Marlowe", Robert A. Logan says:

"These documents** clearly establish both the co-presence of the two dramatists and their attempt to attract the same sources of production for the staging of their plays, thereby increasing the opportunities for influence and creating what several critics have seen as a plausible context for an unfriendly rivalry."

and

"By the early 1590s, Marlowe and Shakespeare were living in London and actively writing for the city’s commercial theater. Born in 1564, a little over two months apart, each had established himself as a playwright by his mid-20s. But, as is well known, Marlowe began composing plays before Shakespeare and rapidly gained recognition and prominence. No evidence has been found to tell us whether Marlowe and Shakespeare ever actually met or whether Shakespeare acted in one of Marlowe’s plays, but both are quite likely. As every student of sixteenth-century drama knows, in 1592 Robert Greene in A Groatsworth of Wit, writing to “those gentlemen his quondam acquaintance, that spend their wits in making plays” [usually identified with Marlowe, Nashe, and Lodge], ridiculed the multitalented Shakespeare, saying, “an upstart crow, beautified with our feathers, that with his Tiger’s heart wrapt in a player’s hide, supposes he is as well able to bombast out a blank verse as the best of you: and being an absolute Iohannes fac totum, is in his own conceit the only Shake-scene in a country.”"

The second quote also makes reference to an insult to Shakespeare by a now largely ignored playwright, Robert Greene.

Widely considered in modern times to be the second most importand playwright and poet of Shakespeare's era, and considered by many to be the greatest during his own life, Ben Jonson had an immense respect for Shakespeare. After Shakespeare's death, he wrote the poem " To the Memory of My Beloved the Author, Mr. William Shakespeare", where he said:

"To draw no envy, Shakespeare, on thy name,

Am I thus ample to thy book and fame;

While I confess thy writings to be such

As neither man nor muse can praise too much;"

(*)Its worth noting that the extent of the influence and rivalry between the two is very much contested, but very few people contest that both occured to at least some extent.

(**) Philip Heneslowe's Diary, which contains information on competing productions of "The Jew of Malta", "Henry VI" parts 1 and 3, and "Edward II".

Sources: "Shakespeare's Marlowe" by Robert A. Logan

" To the Memory of My Beloved the Author, Mr. William Shakespeare" by Ben Jonson

I can think of two examples right off the bat. One of the earliest indications we have of Shakespeare as an up-and-coming playwright is a biting attach on him by playwright Robert Greene (or someone purporting to write as the recently deceased Greene). He takes Shakespeare to task for being an actor borrowing glory from playwrights.

"Yes, trust them not, for there is an upstart crow, beautified with our feathers, that, with his Tygers heart wrapt in a Players hide, supposes he is as well able to bumbast out a blanke verse as the best of you; and being an absolute Johannes Factotum, is in his owne conceit the onely Shake-scene in a countrie." - Greene's Groats-Worth of Wit

Ben Jonson, the playwright who dominated the stage after Shakespeare, elegized the bard upon his death. Check out Jonson's To the Memory of My Beloved the Author, Mr. William Shakespeare.

Note that neither of these would make any sense if applied to Christopher Marlowe, Francis Bacon, Queen Elizabeth, or Edward de Vere.