My personal take : Your first meet should be Neutral space. You do not want to bring a couple / single to your private space till you 100% sure. They/He/She maybe picture perfect inside and out ONLINE, but behind closed doors not always as you thought. Anything can go wrong from The vibe, Chemistry, stalkers ect ect...
So who pays for the Neutral venue ? That would depend on plans. Again personally i do not mind as i prefer staying after or enjoying the entire night and not having to dress up and go. If the Couple plan on staying too an offer to share costs would be a nice gesture as the booking would be upgraded to 3,4 persons.
BUT do not expect 1 side to pay all the costs involved. ( Again pending Scenario/Plan ect )
Sure the safest way is a pre-meet in public space but everything can change behind doors.