I am defiantly from the bring your own camp. I have been to WDW 6 times with kids (2 of them I had 2 kids, 2 of them I was with friends who also had kids the same age as mine.) So 6 trips in 4 years with kids under the age of 4. I know what I am talking about. :lol:
Bring your own, and bring a good one. One that is light weight, (not a full size graco or even flow) folds down good, reclines, has good shade, good storage and trays across the frount that are removeable are good (good for snacks, but come off if the kid wants to get in and out on their own.)
We have taken a back pack stroller for a 9 month old (loved it, Instep.) It was so great. It works as both a back pack and stroller. When the crowds were bad we just converted it to a back pack and walked out with no problems, plus she loved being up high. This is only good from around 6 to 15 months or so. (the major drawback.) It is very small and light too. She was very comfortable in it, and was even able to sleep in it.
A combi travel savy DX (this came on 3 trips and countless others not to WDW.) It folds in on it's self very small, but has all I mentioned before. Very comfortable for the kid. My friend bought the same one. Most take everything out of the basket, so bring a bag too.
With 2 kids we took a peg pergo aira twin. The 1st trip we saw like 4 others the whole trip. The most recent trip (early May) we saw them all over (most of been 50 others.) For a double it does fold down well and is not heavy. Even then we had to put the baby in the bjiorn in order to manage both kids and it on the bus along with bags. We got a pretty good system going.
If you child is a tolder, bring your own. You will want to keep him in it many times (when you aren't in the parks) It is much safer that way too. When he is out of it, it holds the bags well. The rented ones really aren't that nice. Plus you won't have one waiting for the bus, walking too an from, at DTD and around the resort or the airport. All these places can have TONS of walking to do (and napping in the stroller, belive me WDW does wear them out enough for that.)
I might rent one when my kids are much older, but under 4 it is worth it to bring your own.
I never really worry about people stealing them at WDW. I do at the local mall. :lookaroun Don't leave your camera or wallet on it and go on a ride, but the bag is fine.
Oh and did I mention that now the Peg Pergo double ($300) has actually paid for itself for the price of rentals now? All of them did. I paid $100 for the combi and $50 for the back pack one.