Hi Paul -
Rover sitters are encouraged to respond to all requests (the exception being if their profile is set to away mode, in which case you would have been alerted to this before sending a message). A quick response time helps us in the search rankings, so if you're picking someone near the top in the search rankings, chances are good they are diligent about responding. As Karen said, you can view a sitter's average response time and response rate on their profile. If they have a 100% response rate, you can expect a message back. Also, if you message a Rover Pack member (indicated by a badge of a dog in a crown), you're guaranteed a response time within 12 hours. See the picture below for what to look for:
From my profile, you can see that my average response time is 1 hour (not counting overnight) and I respond to all requests within 24 hours. I have a 100% response rate, so I've never ignored a request. I'm also a Rover Pack member, so I'm required to respond to all new requests within 12 hours unless my profile is set to away.
Looking for similar profiles in your area will maximize your chances of getting a quick response to your inquiry. It's also encouraged to reach out to multiple sitters rather than waiting to hear back from the first one if you're interested in multiple. That way if one isn't available, you'll already be in communication with another. You can even meet with multiple sitters so you can pick the one you (and your dog) are most comfortable with. Even if a sitter responds quickly, it doesn't mean your dog will be the best fit with them, so it's always a good idea to have a back up plan.