The specification says that position:fixed elements and their descendants should be repeated on each paged of printed media, and browsers follow this rule.
However, this can lead to content getting clipped out on pages 2+. Try to print this recipe page in Chrome or Firefox, for example, and you'll see that some of the text is overwritten by the repeated position:fixed elements on pages 2+. This problem doesn't occur for non-printed output because this page has a rule applying padding on the root element of the main content to manually move it out of the way of the header.
Given these downsides, is repeating position:fixed on every page a good idea? Perhaps repeating should be an opt-in feature via a new CSS property.