To properly address this, we first need to break down the methods in which one could “release” a play.

There’s a common misconception that Shakespeare predominantly cared for the stage, and the publishing of his plays was merely an afterthought; this line of reasoning is often delivered with the argument that “Shakespeare is to be seen, not read”. Over the past 10 years, Lukas Erne has compiled evidence of Shakespeare’s sustained relationships with printers, the consistent publication of his texts for the first half of his career, and the simple fact that, most published plays by Shakespeare are simply far too large to have ever been acted in full. There would have been almost no reason for him to write plays of 4000+ lines, when actors only had the time to stage around-2500 line productions, unless he sought to consciously cultivate a reputation as a published figure. Shakespeare’s interest in publication can also be seen across some of his introductions to texts, such as Troilus and Cressida which begins “Eternal reader, you have here a new play, never staled / with the stage,”^1 and his poetic interests and publications therein. While some of these lines are more flourish than fact (Troilus and Cressida was probably performed first), it speaks to a man who cared about his print reputation. The notion that Shakespeare cared more about the stage than print can largely trace its roots to the 18th century, in a manner that was never objectively proven, but more guessed as the "proper sort of thing". On the balance of evidence, it’s reasonable for us to say that Shakespeare cared about seeing his plays on the stage, and equally cared about seeing them in print.

With this in mind, the “release” of Shakespeare’s plays becomes a 2-part endeavour. They were first performed on the stage, and typically 2 years later, then sold in print^2 . If we first look at the printing side of things, it’s important to consider that producing texts was by no means cheap. While we’re not entirely certain of the figures, a first edition would have much greater costs associated with its production (Registration fees with Stationers + setting) than subsequent editions. Where an initial printing run of a text could struggle to break even – if a text proved popular enough for a 2nd edition, it could generate significant profits. In this respect, one of the easiest measures of “hype” surrounding an author, would be how often they necessitate re-prints. To quote Erne, in this respect Shakespeare was dominant:

“104 of the 211 plays published between 1583 and 1622, or not quite 50 per cent, reached a second edition within twenty-five years. As for Shakespeare, twenty of his plays were first published during the same years and only three of them […] did not receive a second edition within twenty-five years, so the rate of plays that reached at least one reprint within a quarter-century is an astounding 85 per cent, as opposed to the not-quite 50 per cent for playbooks in general. Of the seventeen Shakespeare plays that were reprinted at least once, seven received two editions, four went through three editions, three through four, Richard II through five, Richard III through six and 1 Henry IV through seven editions. On average, the twenty Shakespeare plays published between 1594 and 1622 received […] almost three editions per play. ” ^3

If you were trying to make money from printing plays, Shakespeare was the closest you would get to a sure-thing. Regarding “PR campaigns” or advertising, one of the most simple methods which Shakespeare’s publishers certainly spear-headed was… simply telling people it was Shakespeare.

Our modern association of plays with playwrights is a device we can largely trace back to occurring alongside Shakespeare’s rise to prominence. The advertising material of earlier plays would often cite the Main actor(s) in the performance, in much the same manner that a Hollywood movie will tell you who the leading man is, but you’ll have to dig a lot deeper to learn who wrote the script. Shakespeare’s earliest plays didn’t often have his name included in the printed runs, but to quote Erne again:

“The proliferation of Shakespeare’s name on title pages from 1598 suggests that it did acquire such marketable recognition value. Shakespeare’s Juliet asks ‘What’s in a name?’, and [printers] seem to have realized that one thing that is in it is money, and that a name to make money with was ‘Shakespeare’ ” ^4

In fact, Shakespeare’s name was so successful, that it was not uncommon for it to be used on plays he had no bearing in. There is also a trend of dramas “Written by W. S.” (such as Thomas Lord Cromwell) which scholarship does not currently attribute to Shakespeare. It’s likely, in these cases, that publishers knew what they were doing, and wanted audiences to assume Shakespeare was the author, while still maintaining some deniability.

For advertising plays across 16th Century London, title pages were the most circulated staple. To quote Erne: ^^^^for ^^^^the ^^^^last ^^^^time ^^^^I ^^^^promise

“title pages served not only as book covers but also as posters, which were put up on doors, walls and posts in London. No other part of an early modern book gives us more immediate access to what London’s publishers and booksellers were counting on to sell their wares. This sheds light on the fact that early modern title pages, including those of playbooks, often praise the book’s content (‘the most excellent history…’, ‘the late and much admired play…’)” ^5

So, to answer your question, when Shakespeare released a printed play, there was enough hype to warrant multiple editions, and a PR campaign that would have involved an extended printing run of title pages, specifically highlighting that it was a Shakespeare play, which were put up around London.

There’s a lot more that could be said about when Shakespeare was staging a play, but to summarise it briefly, it would have likely involved the circulation of playbills in a similar manner. Two of his plays (that we know of) also received longform poems that advertised their contents, of which only the Titus Andronicus ballad survives. While an awareness of ballad circulation is slightly outside of my expertise, I believe these would have been used in a similar manner to playbills and title-pages.

As his plays were being performed, others were being printed and selling through multiple editions. There would have likely been a mutually-beneficial element to this as well, between his name being used to describe high-selling texts while simultaneously debuting new material in the theatre. In short, telling people a text was by Shakespeare was sufficient for generating significant hype.

References

1 - https://shakespeare.folger.edu/shakespeares-works/troilus-and-cressida/preface/

2 - Lukas Erne, Shakespeare as Literary Dramatist, Chp 3.

3 - Lukas Erne, Shakespeare and the Book Trade , Chp 1

4 - ibid, Chp 2

5 - ibid, Chp 3

^^Apologies ^^for ^^the ^^lack ^^of ^^page ^^numbers, ^^my ^^digital ^^edition ^^doesn't ^^have ^^them