I am not a lawyer. Consult your local laws for details on possible jail sentences. My experience is that actual court data is notoriously unsearchable for a layperson, so I suspect median and mean sentences would be very difficult to calculate. These are also modified by judges who reduce or dismiss the charge.
My understanding (IANAL!) is that the probability of being registered as a sex offender, assuming you've committed a crime on The Big List Of Sex Crimes, is 100%. Consult your local laws for details on whether public urination is actually on said list, as it varies from jurisdiction to jurisdiction.
You also have to factor in whether the police officer actually cares enough to arrest you (which, if you're clearly a sober, polite, middle-class, white, male, is probably fairly low)
My local laws are probably very different from those valid in Alaska. I am asking just for curiosity, since it hadn't occurred to me that urinating in public can be such a major problem. I am not planning to travel to Alaska to urinate there.
link: http://www.philosophersbeard.org/2012/07/why-prison-doesnt-work-and-what-to-do.html
I came across this short well-reasoned essay via reddit, but most of the discussion there was by people who (ironically enough) appeared not to have read it. The comment section under the article itself was similar. Some seemed not to have read the first paragraph.
The author's arguments against the imprisonment system, and for alternative methods of punishment, are interesting. They touch upon ideas which are hard to consider rationally. The very idea of punishment is something most people appear to find unpleasant. Indeed so it must be if something is to be considered a punishment.
I found some relevant previous discussion on Less Wrong (Crime and Punishment, The Wrath of Kahneman), and Overcoming Bias (Prison is Cruel), but these seem to be about specifics, and not the system in general.
I am curious both what the scientific consensus is on punishment systems, and what Less Wrong thinks of them. With such an emotionally charged issue, it's hard to find rational discussion about it.