Client wants to do a meet and great with out their pet?

You could try asking why they don't want to bring the dog and let them know this is a good time for you to get to know the dog's personality and even mention it is a requirement if you want. If you are concerned for your safety, definitely start off in a public place and if it goes well then you can always set up another Meet & Greet at their house or yours depending on where the service is going to be.

Sometimes people just want to interview the sitter to make sure the sitter is legit and it can come off sounding odd. If you are really concerned, you can also always contact Rover support and have them take a look at the account to see if anything weird is going on.

You never have to do anything that makes you uncomfortable.

I always require that they bring the dog to a M&G. For me, the purpose of the meeting is to see how the dog acts in my house,how it reacts to me, for me to interview the owners and determine the dog's schedule and any special circumstances, and for them to meet and interview me. I will not accept a booking without meeting the dog first.

I would ask why they don't want to bring the dog and explain to them that you need to meet it to see how you interact.

NO WAY! you're doing the stay or visit with the dog...the dog needs to be there lol otherwise, it seems creepy to me.