The other respondents answers are excellent. I’ve done both, and either can be lucrative. If you are confident in your ability to attract clients, and don’t mind changing locations between every session, mobile/in-home training is for you. It allows you to set your own rates without any overhead. Do keep Debi’s response in mind, of course; you’ll need to factor in fuel and vehicle costs.
On the other hand, if you prefer not to carry all of your tools with you, a gym can be lucrative as well. You can see more clients in less time, and most gyms have fitness and/or membership directors who facilitate sending clients to the training staff. Since you’ll be sharing profits with the gym, you may need to get a bit creative to earn more per hour. Seeing two or three clients at once is a great option, in or outside the gym, allowing you to make more per hour, and maximizing your potential referrals.