My specialty is music and I can do music for games. However, if I were to get a job at a bigger game studio per say, would I have to also make sfx cause I have no experience in that...

It really depends on the studio. Some have composers that do both, and others split the duties up between different people. You always do just the music and "Bob" always does all of the sound design.

Major studios like Playstation and such more than likely split the duties up. One of the best ways to tell is to pay attention to how they market their job openings. If they post a job saying "Sound designer position" and never mention anything about composing in the entire ad, you can pretty much assume they divide up the duties.

At my job I have to do both- which was a stretch for me. While I had done SFX for many games, sound design is an art entirely unto itself and I had a good deal of learning to do- still do. But with practice, some research and a good ear- you can learn it.

I hope that helps!