How can i be assured that the sitter whom I have my dog staying in their home for daycare or boarding, verified vaccinations of the other dogs staying with them?

You would have to discuss it with the sitter and ascertain what requirements and documentation he or she requires. You should also remember that legal vaccination requirements vary by state so there is no Rover mandate on how to handle this issue. Each sitter sets his/her own requirements and procedures. When you are interviewing potential sitters, you should find out how they handle their charges' medical documentation. However, for your own peace of mind, perhaps you should be searching for sitters who only take a one dog at a time.

When you came to them, did they ask to see proof without you asking? If so, that is probably their common practice. If not, I would flat out ask what their policy is. I would ask "Do you ask to see proof of vaccinations and do you have proof of vaccinations for your own dogs?" Any sitter should be able to answer that immediately (ask face to face in meet and greet).