Have they had to pay more on average in salaries because private sector competitors (including private universities) now know exactly how much to offer someone to entice them?
I've occasionally heard of these lists being used to identify woefully underpaid faculty members who might therefore be disgruntled and easier to recruit. However, I don't think this has had a substantial effect overall on salaries at public universities. Most faculty members don't inspire bidding wars between universities, and recruitment is based on many other factors beyond just salary.
For context, note that average salaries vary substantially between universities, and these differences are sometimes pretty widely discussed in the community. Even in the absence of data on individuals, that's potentially useful information for recruitment. However, there seems to be no trend towards salaries evening out. Instead, they tend to end up balanced with factors like desirability of location.
Have they found it more difficult to recruit people from the private sector (including private universities) who might not want their financial details to be public?
Not to a noticeable extent, at least for ordinary faculty positions. This could be more of a factor for mid-level administrators, whose salaries might come under greater scrutiny. (By contrast, the salaries of top administrators are public information for every non-profit university in the U.S., public or private, because they must be reported on IRS Form 990.)
Have there been any other negative or positive side effects?
Transparency about salaries has a weird mixture of effects. On the one hand, it makes the overall patterns clearer, and anyone can judge for themselves whether the results are fair. For example, it's easy to gather data on whether women are being paid less, whether there's salary compression or inversion, whether different people receive comparable salary increases upon achieving tenure, etc. I don't know of any formal studies (which could be interesting), but there's at least a fairly widespread belief that this transparency helps cut down on abuses.
On the other hand, it can also increase disgruntlement. The actual salaries are almost guaranteed not to align perfectly with what would seem just to any given person (because of course different people have different visions of what would be appropriate). I've certainly looked on occasion at salary lists and wondered why on earth X was being paid 15% more than Y, and I once talked with a friend who had discovered that he was Y in such a case.