I agree with Walt. The profiles are important for you, as a sitter, to make sure you have updated information, know if the dogs are good with other animals, etc. Many pet parents go onto Rover to browse for a sitter, without taking the time to create the full profile before sending requests.
If I'm able to schedule a M&G prior, I typically say something like: "Great, I'll see you then! In the meantime, please complete your pup's Rover profile so that I can look it over before we meet. That way, I can ask the right questions and make sure that I can have easy access to their care instructions during the stay!"
If I can't meet prior, I have very strict guidelines about what kind of dogs I will allow into my home. Generally, they have to be referred by an existing client or they must consent to a veterinary reference check. If neither is an option, they must make time to come to my house so that I can see their dog's temperament before booking. It generally goes something like "The meet and greets are really invaluable, as they help a dog acclimate faster with less stress. I understand that time might not allow for one on such quick notice. In lieu of that, can you please take a minute to fill out your pup's profile on Rover so that I can make sure it's going to be a good fit? I also require a vet reference for new clients that can't do a meet and greet, so please make sure their contact information is filled out. I just need confirmation that your pup is UTD on all shots, etc!".
One strategy to avoid last minute requests - always mark your calendar as unavailable for the next few days or so. I am new to Rover and just had a last minute booking with no M&G. I interviewed extensively on phone, felt comfortable and took the dog. All went well, and I have a great client now!