As I understand it, the situation is this: copyright in the images belongs to the Bodleian, and use of them must be properly licensed.  They have no objection to hyperlinks direct to images, though obviously courtesy demands that the source be specified; I always try to provide an additional link to their homepage.  The status of textual content is less clear, but I see no reason to suppose that transcriptions are a problem, though clearly they need to be made with care, and any uncertain readings indicated as such.

It is worth mentioning that some examples are difficult to read, and in these cases you really do need proper image manipulating software which will allow you to zoom in, adjust contrast and so on.  Older examples in particular often employ symbols and abbreviations which are now rarely used; in those cases some additional background knowledge is required.  Detailed information on the copyright situation is available at the Bodleian site: Copyright & Bodleian Images.

My personal feeling is that any attempt at wholesale posting of trancriptions here would be a bad idea; broadsides very often do not specify identifiable melodies, and we already get far too many songs without tunes posted.  However, there are certainly times when a broadside text can add worthwhile information to a particular discussion; I usually just provide a link direct to the image, but an accurate transcription accompanying such a link might be very helpful for people who are able to view them only via a browser.  Of course, I speak for myself only; Joe Offer would be the best person to advise.

As to whether the Bodleian has any interest in the kind of (inevitably amateur) transcriptions that you or I are able to make, I have no idea, though I rather doubt it; you could always contact the project director, Mike Heaney, via the website (Project page).

Malcolm