The playbill seems to be an odd responsibility of the stage manager and indeed not all of it is. The parts of the Playbill that the stage manager is responsible for will vary slightly according to the kind of company you are employed in. At the very least you will be responsible for making sure the casts name's are spelled correctly. You will give them an opportunity to approve their bio's if they get one. You have confirm that show order, musical number order, scene order, Act definitions, etc. You will confirm any special thanks that you have been made aware of and special mentions should your show require them. You will confirm the artistic and design teams names and order in the playbill along with their bios and director/dramaturg notes as necessary.
Perhaps the most important thing of all when working with the playbill is the process that you all follow in keeping track of the changes as they are made and what 'edition' of the playbill you are working with.
In my earlier days we got a "blue line" copy of the playbill (that copy that had graphic lines on it for margins, etc). We carried that to rehearsal with us. Each cast member, designer, and member of the artistic team got the opportunity to read their bio's and initial if correct or make corrections. Along with the director we would go through title page, scene breakdowns, etc. to insure that they were accurate to what we knew the show to be. The 'initialed copy would go back for edits and corrections.
In many cases you will be included in the synopsis of the show, provide a list of the cast and roles, a chronological listing of the scenes and musical numbers. The different shops may request that special thanks and consideration be given to individuals and companies who may have provided services at little or no cost.
A new dated blue line would be reissued with the warning that this was the last one to make corrections and to please make sure ALL spots have been initialed. It would include the first blue line so that previous initialed responses were included.
The final initialed copy would be returned to the publicity department for publication as the Playbill for the show.
Included, of course are many aspects of the Playbill we are not responsible for; advertising, number of pages, page order, etc. But you and the director should carefully go through the show information for accuracy and consistency.
There are many contractual issues that the stage manager is not responsible for. Contract riders that list exactly where names appear and in what size type are not typical responsibilities of the stage manager.
Union requires notification in the Playbill of all union members and generally on the title page. You should check the appropriate area of the contract you are working on to insure you are meeting union obligations to the union members as it regards to the Playbill.
The order of designers is important and historical. When you list the designers make sure it follows the correct order. Scenic Designer, Costume Designer, Lighting Designer, Sound Designer, Stage Manager, and if appropriate, Stage Manager.
Most type sizes are in relationship to the title and you will see language that refers to the "percentage of the title". Authors and composers commonly get what is called 50% of the title meaning that their names are half as large as the title on the page. Some people such as producers and stars are "above title" while most of the production team and actors are "below title". Directors float around a lot but are either the first listing below the authors and composers or as it seems to be favored these days at the very bottom in 50% of the title type.
In the 21st century there are numerous ways to distribute a document for 'editing and reviewing'. I do not advocate one or the other. I can tell you from personal experience that getting the hard copy and having the actor 'initial' the document saves loads of heartache when questioned about who did what when. A proper process that lets you know the date and iteration of the playbill is the most important consideration for successfully negotiating through the ego web of personal and biographical information.