I'd be extremely wary of watered-down shenanigans: if you're hoping to "play along" with a sick call in order to avoid liability, you will find that it doesn't actually work that way: if you knew they were doing it fraudulently, you're still responsible for allowing and encouraging them to skip off work. Likewise, if you give a brief speech at a company meeting ("Look, I know you're doing this, and I need to report it if I know about it, so please don't do it in front of me."), this similarly doesn't actually protect you from responsibility (in the ethical and legal senses) for what follows.
What I would do is speak to the producer. "There have been a number of requests, and I think allowing actors to take days off would be good for company morale. It would also help to keep the understudies fresh. Of course, we need to make sure this is handled responsibly, but people have been grumbling and talking about calling in sick in order to do other gigs, and I think we need to nip it in the bud."
In terms of scheduling, you do what you always do: schedule as best you can considering all the information available to you. If you don't think someone will be present, don't schedule anything which requires their attendance. If you have ambiguous information, call them and give a straight question: "Should I schedule you to be present on Monday, or will you be occupied with other things that day?" If they lie to you or mislead you or refuse to commit, then it's their own fault if the schedule results in their missing a call or being sent home for lack of need.