If you already have a following and do not anticipate ever using your web site or social media to get new clients you could probably do that.
However, generally I agree with karin…. when people are looking for a trainer they often start with an online search. And even if they are given a recommendation by someone they know they will look up the place/person online… not giving an address puts you in the position of missing out on business, makes it hard for people to locate you (sometimes I need to go someplace and cannot find the business card I was given and just look it up on my smart phone to remind myself of the address. Actually I do that numerous times a week. And for people who do not know you personally it can send a message that there is something questionable about the business.