Does my kid have to take skating lessons at a figure skating club or can she get good training at the community centre?

Many community skating rinks offer various learn-to-skate programs. These can be very hit and miss. Some of these programs have good coaches and the kids in the program may be fairly focused on the sport, but often you're getting the basics. You aren't choosing a performance kind of environment. In general, if you want the best coaching and highest probability of long-term success, a skating club is the way to go. And yes, Skating Clubs typically cost you more than a community rink learn-to-skate program.

This is definitely a you-get-what-you-pay-for situation. I've watched kids who come to the club after spending a year or two in a community program and while they've learned the essentials, they typically haven't reached the same performance level of kids who have been trained at the club. As an example, the club skaters typically skate backwards much better than the community rink skaters. But, with some good coaching and some work by the kids, they do seem to catch up so some time at the community rink may not be a career ending move. However, I would recommend that you at least take the time to observe a lesson at both locations before you make a decision.