pros and cons of one's own papers covered by media One published a paper on a journal. The communications team of their school want to publicize that paper in media. Is there any substantial pros and cons for the paper writer?
The main effect of media coverage is increased visibility. In case of very controversial research that might be considered a con, as it might invite harassment or people running with unsubstantiated and possibly dangerous claims. In most cases, however, I'd see it as a pro. You get to communicate about your research to groups that would otherwise not have seen it, effectively doing a form of outreach and educating the public on cutting-edge research.

Other pros:


It can be fun - especially if well-received. 
It can be good practice in communicating to non-experts.
It improves your professional visibility as well as the visibility of the specific research project, and the visibility of your institution. It's hard to measure this effect I suppose, but it might produce new opportunities to e.g. give an invited talk somewhere, or lead to students being interested in working with you.
Your employer or funding agency might also like seeing this kind of non-citation impact.
At least some US institutions would consider media coverage of your research as evidence of service (along with committee service, traditional outreach and so on) for tenure evaluations. One big reason to have this factor in to the decision is that such activities help promote public awareness of the work that takes place there.
Some members of the audience might find the research an appealing answer to "why are we funding these people with our taxes again?" in a way that wouldn't happen if we only ever communicate with other experts through pay-walled journals.


Other cons:


It does take some time. (It's almost always a negligible amount compared to the time you spent on the research in the first place, but busy people are busy.)
Misleading reporting could be hard to deal with in some cases. If you work with a university or journal press office on a press release you can usually make sure that the release itself is accurate, but then a game of telephone may ensue as different outlets pick it up. The risk seems to be higher the more impactful the result, as the media coverage after the press conference about the OPERA neutrino anomaly clearly showed, or if the research is politically controversial - e.g. climate change.
Attempting to maximize coverage could lead to overhyping the research, which eventually erodes trust in academics as a whole.


So overall, I think the only real con for most regular-stakes research is that it could use up a single digit number of hours. Hence, I'd suggest trying it at least once. For 
 research that's potentially more controversial it would make sense to first discuss the likelihood of adverse consequences with a more experienced colleague in your field.