There's a few other things to consider -- what will happen when you do teach at their house. When I lived in Florida I decided to agree to teaching at a student's house as it was sort of on my way home from work. A few other families in this rich, gated community heard I was coming in to do a lesson and also hired me. I didn't charge them an outrageous amount as it was on my way home. I didn't forsee the problems I had to deal with despite having a syllabus and being forthright about my expectations. The kids all really enjoyed lessons with me and we had a lot of fun, but the problems made me give up these lessons after two years.
* Students are not as well behaved or focused in their own home. My students would typically just walk away from the piano to answer their parents, and the parents wouldn't think twice about calling the child away from the piano in the middle of a lesson. I dealt with each issue as it came up, but the parents apologized half-heartedly and brushed it off. It happened again of course. The solution is having the lesson in a space that is not high traffic and that is not their bedroom (with all the toy/game distractions)
* Parents are also less respectful when you teach in their home, as I mentioned above. Parents tended to be very casual when lessons started late because they were just busy doing something else in the house, and would expect me to stay and continue teaching. I didn't, but the problem was always there with each family (except one...coincidence that they were one of the few "middle class" families living in this gated community?)
* You'll have to deal with more gifts, more invitations to dinner, offers of glasses of wine, etc, etc. Some of these things were the parents actually trying to get me to stay longer, or teach beyond when I decided to stop. It was very strange and uncomfortable. Things that would never have happenned if lessons were in my own controlled environment. I couldn't decline cute gifts from the children but did turn away many dinner invitations and glasses of wine because I didn't feel it was appropriate.
All in all, I was very sorry to have taught at their houses. The children, in general, were the most unfocused bunch I've ever taught, retained the least, and adopted the attitude that I was there to serve them; there was little respect. I teach out of my house these days and 99% of my students study hard, are very respectful as are their parents. I enjoy it a great deal. If you need the money, as others have said, make sure you charge them something that will make them really consider how convenient piano lessons are or aren't. I agree with you that driving to them in the winter is silly -- it's a risk you don't really need to take.
I question their actions and motivation. You ought to talk to your teacher and tell them what's up. You could risk offending your teacher by taking on these students and making your relationship tense (or tenser). I don't like to take students I know have been sneaking around behind a teacher's back and havn't been honest with them. I know that they'll just end up doing it to me as well. If they haven't talked to their former (your current) teacher about switching teachers, stopping lessons, and asking you to teach, I'd be wary.