Technically, yes. You CAN charge for a meet and greet by setting up a drop-in visit. But realistically...the market isn't on your side. No other sitters are likely charging for a meet and greet. The whole purpose of the meet and greet is to ensure that both parties are comfortable with the arrangement before money is exchanged.
Like Walt said, definitely look into getting your own insurance.
Second, there are other things you can do to limit the odds of something bad happening at a meet and greet:
-Don't allow children to attend meet and greets. Client children or your own (if you have any). I tell clients children under 13 are not allowed on site for liability reasons.
-Never let M&G dogs interact with your guests. Prior to the meet and greet, you don't know if the dog is reactive, aggressive, snaps out of fear, etc. Don't let an untested, uninsured dog interact with paying customers. I always keep boarding/daycare clients in a bedroom or in the backyard while a meet and greet proceeds.
-Ask the owner to arrive with the dog leashed so that you it is under the control of its owner. Once you meet the dog and have a sense of its anxiety level, you can proceed to do leashed introductions with your dog(s) if you have any.
-Allow the dog to sniff around before introducing other dogs to the mix so that they can settle down a bit before being overwhelmed with a new pup to meet.
-If you host multiple dogs at a time, ask if the dog is used to being around multiple dogs in an off leash setting (dog parks, daycare, etc). If not, be extra cautious with introducing them to the high stimulation scenario of multiple dogs. Book a test day of daycare/overnight or ask to meet at a dog park so you can see how the dog does in a group situation.