I'm not a "child person," but there are quite a few in my program, so I'll offer what I've been able to gather on the topic. As far as I know, and as others have already said, the number of schools offering concentrations in child psychology is much smaller than the number of programs that will allow you to work primarily with children.
In choosing programs of interest, and again as others have said, focus on identifying those departments with faculty who predominantly work (clinically and research-wise) with children. You'll also want to select schools whose practicum placements include sites at which children would be an available population. As with my specialty (neuropsychology), whether or not the program explicitly offers a concentration in "child psychology" is much less important than whether or not you'd be able to focus on your interests during the course of obtaining a general clinical psychology degree. Furthermore, I'd make the argument that such generalist training tends to make you more competitive for internship, and offers you a more solid foundation of knowledge upon which you can then build your chosen specialization.
I will agree that where you intern, and even more so where/if you obtain your post-doc, affects your specialty training and status significantly more than your doctoral program. However, in order to get into a child-focused internship, you're likely going to need a good amount of experience working with children in order to be competitive with other applicants.
Finally, in terms of job outlook, from what I've heard, the overall prospects for clinicians working with children are better than for those working with adults. This is owing largely to the fact that there are significantly fewer child-focused clinicians being churned out. Child neuropsychologists in particular are, again from what I've heard, in rather high demand.
