How do selection committees ensure that they won't steal an applicant's ideas as presented in his/her research statement?
Academics are presented with confidential material all the time. We review manuscripts and grants and we are expected to not steal those ideas. A research statement is very similar to a grant proposal. When we are given applications to review we are reminded that they are confidential and should not be circulated. This is the same as when reviewing a grant. While many funding agencies and publishers have clear statements that reviewers must agree to, departments generally do not produce such guidelines. That said, a department would likely come down very hard on an individual who did not respect the confidentiality of an applicant.

With that out of the way, it is often that a reviewer has similar ideas as an applicant. On the face of it, it may appear that the reviewer has stolen the idea from a proposal, when in fact it was an independent idea that the reviewer had already had. One question is should reviewers decline to review based on a conflict of interest when the proposed work is similar to their current/future work. In my opinion, they should not since they are in the best position to review the work and the similarity is likely the reason they have been asked to review. This then does leave the burden on the reviewers to be able to document that the research agenda is in fact their own and not stolen from the confidential proposal.

The final issue is to document anything that is unique to your proposal. Saying you want to study topic X does not make it yours. Saying you want to study topic X with method Y, isn't much better. If the proposal says I want to study topic X with novel method Y because of reason Z, then the linking of X, Y, and Z might be important. If you document this, then at least if someone studies X with Y, you can get the credit for coming up with the brilliant idea of using method Y on problem X. In some ways realising that problem X can be solved with method Y is more important than actually solving the problem.