If I refereed a paper at one journal (and it was rejected) should I referee it again when it's sent to another journal?
There are no problems associated with you reviewing a paper again, regardless if it resubmitted to the same or a different journal. Since you have seen the paper earlier and know its earlier problems you can better judge how it has improved and to what extent it is now publishable. You should definitely let the editor know that you have been involved in the process of this paper earlier even if he/she has not indicated the paper has been rejected somewhere before. That information provides the editor with a better perspective of the development of the paper and can make the decision to accept or reject easier since the willingness or capability of the author(s) to improve the manuscript is set in perspective.

You can always contact the editor and state that you are willing to take on the review but felt it was necessary to convey the information. I do not see it as necessary but it is of course a nice gesture.