Our stroller was never bothered when we were there in January. However, a lady I met on the bus said her was stolen, but was found and her daughter's sweater was missing, but it was otherwise in the same condition. She told me that someone took her stroller because they could not find her own and it was located in 2 days because the theif's stroller turned up with a cell phone number on it. When the CM called the number, she basically confessed to taking a replacement when hers was moved. (In some of the larger stroller parking areas, the CMs arrange strollers to make room for more.) Disney CM arranged for her to drop the stolen stroller for its rightful owner and she picked hers up anonymously as well. The people who were without a stroller were given a rental free of charge. She did say that the CMs told her most strollers are recovered and not usually stolen on purpose, mostly mistaken for their own. I would suggest to tie something to your handle so it can be ditinguished among the crowd. I think that most of the time it is safe to leave your stroller, but nothing is ever 100% safe with the number of people in WDW.