Christopher Barker (and Richard Jugge before him) put out a lot of edicts that start with "By the Queenes Commaundement" and touch on all sorts of things, particularly public health measures around preventing the plague.
In 1578, for instance, Barker's press puts out Orders thought meete, which lists a bunch of orders for city administrators, then a bunch of plague remedies for physicians. Would the poorer people ever see these Orders, or would they rely on city officials and physicians to disseminate the information?
During the outbreak of 1592-3, multiple documents are printed on restricting access to the court, or adjourning the legal term and moving the court away from London due to widespread contagion. In this example, where would this document be posted? Would there be checkpoints two miles outside the court with a copy of this? Would churches or marketplaces display them? Or are they explicitly for city administrators, who then pass the information down to the people?
If they were posted someplace, the big question is: could everyone read the document? Or would they require the use of someone to explain it to them?
The items you are discussing are normally called "proclamations," and they are a rich area of study. Their basis in law comes from the 1539 Statute of Proclamations (Printed here), where the monarch's official proclamations are to be obeyed as though they had the force of a statute, mainly for specific purposes including plague (as you mention), religion, rebellion and trade. The best resources for this are the Tudor Royal Proclamations edited by Hughes and Larkin; Youngs, The Proclamations of the Tudor Queens; and for quick free reference, Steele's Bibliography of Royal Proclamations of the Tudor and Stuart Sovereigns.
Next, the proclamations were printed, but were sent out to the sheriffs or Lords Lieutenant in the counties. These men as representatives of the Queen would literally "proclaim" the text of the proclamation to audiences, either gathered at the church, market, assizes or quarter sessions (the latter two are the Queen's law courts that met in the counties). The documents themselves could be posted near anywhere, but this was less common than in the seventeenth century. In the cities, there might be literate people to read them in taverns or inns, but the people would be more likely to rely on the information by word of mouth (Fox, Oral and Literate Culture in England). In the countryside, it makes no sense to post something when the population density is too small to have a substantial number of people to hear them. When the Queen's courts (law courts) assembled, a large portion of the locality would show up, even if they did not have a case to dispute. While we credit the sixteenth century with the rise of literacy and the printed word, that did not take away from the spoken word. Oral and print cultures coexisted within English society quite comfortably.
And lastly, on Barker specifically, it's worth noting that his printing of proclamations was small business compared with his biggest ticket item. In 1578, Barker was granted the exclusive privilege to print the Bible in English (ODNB Online, "Christopher Barker," by David Kathman). For this privilege he paid 3000 pounds, but it was a lucrative business as under Elizabeth, every one of the nearly 9,000 parishes in England had to own an officially sanctioned Bible. Barker also owned six printing presses at the time of his retirement in 1588-1589. Checking the Short Title Catalog, Barker's presses were responsible for at least 336 different impressions over the course of his 15 year career. While many of these were the proclamations you mention, they were but a small part of his massive printing empire.