I recommend getting a bike in Europe. Unless you have a very special bike that you simply must use, there are plenty of bikes in Europe of all types and all price ranges.

Getting your bike to/from Europe is a hassle. If you plan to check it in as luggage, you will pay through the wazoo. I just returned to the USA from Europe last week and KLM/Northwest checked every bag to make sure it did not exceed the weight limit of 32 kg.

I often tour overseas. Once I brought a bike with me, but only once. It didn't make sense after that.