I loved renting. My kids were old enough to walk until we got to the rental area though and I just found it wayyyy easier carrying them onto busses monorails and boats to sit in my lap rather than stuggling with foldng up a stroller and unfolding it as soon as I got there.
My kids thought it was comfortable (I didn't have an infant that needed to recline) and they just took turns getting in it as they were tired of walking. the youngest was 2 and would put his stuffed dog on the seat and lay across the seat of the double to nap when he was really tired.
I thought renting a double made our trip way easier. I took a single umbrella for the airport for the 2 year old. We did end up taking this to the parks for the 2 year old toward the end of our week because the 5 and 6 year olds were doing much better at the parks with a stroller (they lasted longer).
I never had a problem with anyone taking my stoller. The CM's do move them constantly to clear walkways etc. so it helps if you tie a colorful scarf on the handle to easily spot it in the sea of strollers. Sometimes we had to search for our stroller a little bit as even if you put in in the designated stroller parking area outside of a ride, the CM's might be compacting the space to make room for more strollers and it will still be moved. This still happened with our personal stroller, so that should not sway you.
I think it depends on age of kids and where you are staying. If you are at a monorail resort or boat access...I would take what you have, because it can be rolled on to the trasportation with out folding...if you need to take a bus to get around....and you don't have an infant, I would rent.