So I know that the owners usually contact several sitters at a time. If the owner has already agreed with someone else, am I going to know if the owner doesn't tell me?
No, there is no way to know. But a good way to go above and beyond is to check back with them in a few days. I always drop requesters a line before I archive them, just letting them know I'm thinking of them and ask them if they were able to find a sitter and that I'm still here if they need anything.
Even if they did book with someone else, that can go a long way in the future if that sitter doesn't work out or becomes unavailable. You never know when that relationship will work in your favor!
Like Jennifer mentioned, there's no way to know how many they've contacted. I always try to respond immediately, and suggest a meet and greet for as soon as possible. They're more likely to book with the person they meet with first, assuming it goes well. At the M&G, I generally ask if they are ready to book. If they want to wait, I point out how great it is that they are taking the time to find a good match for their pup (makes it less awkward for them),and then I remind them that I can't hold their spot unless the stay is booked. So, I generally suggest a time for me to check back in. This lets them know you're a good sport and are genuinely interested in creating a relationship with your clients. "I'll touch base with you early next week, and we can go from there. If my schedule fills up in the meantime, I will definitely let you know"