You might scoff at the very notion that there is an effect, unless you have an understanding of economics. I assure you that what follows is not just because I work in government, but it is in the spirit of good analysis.
It is useful to think of the labor market as being two discrete parts: a public sector and a private sector. This is a gross oversimplification as there are sub-sectors in each that are more numerous than a blog post can readily accommodate as a forum for discussion, but it will serve our purposes. The demand schedules for labor in both are driven by the demand for their services as well as the ability of both to compensate workers for those services. Nothing earth shattering there. Their supply schedules, too are fairly similar, and they are functions of each other to a certain extent.
To explain that point a little more clearly, it can be fairly reasonably said that if public sector compensation is increased, all other things being held constant, that workers at the margin will go to the public sector instead of the private sector. In that shift, what you will see is that the marginal workers that shift from the private sector to the public sector are of a talent level that the private sector would otherwise liked to have had, assuming both markets were in equilibrium previously. In order to secure the same quality of labor that the private sector had previously, private firms, on average, will increase wages to drive a shift in labor supply back to the private sector. Conversely, a decline in government compensation that is determined exogenously (by elected officials, for example) will shift the labor supply for government in while shifting the labor supply for the private sector out, which reduces wages in the private sector.
The illustration of this is fairly simple in the event of an exogenous reduction in public sector compensation (CLICK ON PICTURE FOR LARGER IMAGE):
As I said at the outset, this is fairly over-simplified as there are a lot of little nuances here and there. For instance, many in the public sector are trained in fields that do not have, in the terms of private sector human resources advisors, "transferable" skills to the private sector. For instance, fire fighters and teachers both would have a difficult time finding equivalent private sector jobs, though teachers would find a narrow number of opportunities in private sector schools. As such, many public sector workers may have a difficult time switching between the two sectors.
The long and the short of it is that the recent efforts to roll back public sector compensation at nearly all levels will likely further add to the deflationary tendencies in the labor markets that exist right now.