Strollers, and items in strollers (and on benches), are taking purposely by people.
However, it's not the norm.
I would take your own, make sure you don't leave anything in it. Park it "backwards", so if handles are facing you pushing it in, turn it around and park it so it's awkward to get to it. Be sure to use the brake.
Some suggest getting a small cable lock and locking the two BACK wheels together, so someone taking it away would have to carry it (awkward and obvious) or tip it forward to push it away (obvious and sillyloooking). You do not want to lock front and back wheels together, nor do you want to lock it TO something, because the CMs have to be able to move the stroller for their own traffic purposes. Locking the back wheels allows them to move it.
Remember, the main time that strollers become necessary is when you're on your way back to the hotel. If you're staying offsite, that 10 minute walk can become rather long when carrying a big kid or coaxing a bigger (but still small enough for a stroller to be an option) kid to walk. And that's just when you have to turn in the rentals (sad).
I have heard people say that you can get your rental and have them put your own stroller in the back somewhere, but I know nothing other than "I heard about it on the dis"...that would be the best of both worlds, since ou have the Disney visa.