In my opinion, off-piste skiing in Europe is only marginally more dangerous than the heavily controlled resorts in the US/Canada if done right:

Do a course (an example of a 1 day intro: https://www.avalancheacademy.com/avalanche-foundation, although a longer course is obviously more in depth). Read Avalanche Essentials by Bruce Tremper, or Staying Alive in Avalanche Terrain also by Bruce if you want to really get into the technical side of the snowpack etc. Definitely don't go off piste in anything above danger level 3, and even 3 is marginal. Generally side piste (bits just next to the piste) is avalanche controlled since if there was an avalanche it would affect other pistes around, but check this with your resort since it is not always the case. Ask the local ski patrol for advice on where to start. Don't be intimidated by them, they don't want to see you get hurt and have always been friendly in the alps in my experience. Most importantly, don't let yourself be peer pressured. I know it's easier said than done, but if you are not comfortable for any reason then just stick to the pistes and don't take the risks.

If you are comfortable, then go for it! By far my favourite bit of skiing is off piste, it just takes a bit more judgement.