We took our kids to Shenzhen for a few weeks a Summer ago and enrolled them in a private international school up there in the hopes they would pick up some Mandarin. It did not go well. The students already had a cohesive group and we wary of a new person, the teaching style was significantly different from what your children may be accustomed to, the facilities while 'nice' were not quite upto the standards you'd find in Hong Kong. Generally the experience did very little to help our kids Mandarin. In fact, it probably hurt as it showed how behind they were in that language vs. their counterparts up there who speak nothing but Mandarin. Our kids self-confidence took a bit of a blow.
Last Summer we opted instead of a Mandarin day-camp at their HK international school. This was much better in that our child was with people he knew (or saw around the school) and he was in an environment he was familiar with. He ended up improving a great deal, and was speaking it at home more often than anytime in the past. He's gone into school this year feeling much more at ease than he was last year, when he was very nervous.
There are some places that cater to international students in Shanghai and I've heard good things about a few in Taiwan. Not sure what we'll do next Summer. One of the things we discovered was his own self-confidence was very important to him using a second language, and being in an environment where he was constantly reminded that he wasn't a native speaker was problematic.