In my place I would guess its around 50-50
I am a drummer and studio owner. I have a wonderful vintage Ludwig kit that's always tuned up and many people are happy to play on it. Eager in fact. Sometimes a guy will bring his drums, but leave them in the car after he checks out the house kit.
The pros know what drums are in a studio already and besides, the more a cat works, the more he likes to travel light.
But as a drummer, I understand how it sucks to be _forced to use someone else's instrument. Imagine a bebop jazz guy sitting down to a drum kit set up for metal or vice versa.
I get a lot of different styles in here and I suppose very few clueless young bands. I respect the musicians who come here and _assume they understand their instrument as well as I understand mine.
Nobody would even think of demanding that guitarists coming in use the 'house guitar'. Leave your Les Paul at home- we have a special Strat that is the official Studio Guitar. Back when I did record kid bands, I saw as many crappy guitars and amps as I saw crappy drum sets. It all goes back to the Rodney Dangerfield Principle as applied to drums- no respect.
At a jazz rehearsal a few years ago, we were trading fours around the room. I (the drummer) assumed it was my turn and I took four and the conductor said "no! just instruments!"