Here's my problem, and it's not just about the building: they're intending the building to be a for-profit venture, with which the county can "make a few bucks on the side" -- if there were enough profit, you could imagine them lowering taxes or providing other services, because money's flowing in. More commonly, you'll find yourself paying fees for various government services -- registering a car, getting something assessed or inspected, and other such things. You'll pay the government to provide you with copies of a document (they get pretty hefty, and reams of paper aren't free) or to search a database for something at your request.
Fee-for-service is common. Imagine a world where we didn't force people to pay for the services they are provided with: some jerk would come in every few days abusing the system, forcing everyone else to wait, and he would do so with impunity. "Run that search for me again, just in case." "Can I have another copy of that 4000-page document? The last one got a stain on at least one page, I threw the whole thing out." Forcing people to pay a fee keeps the cost of abusing the system high, discouraging anyone but rich people with nothing better to do from doing so. The other nice thing about it is that you don't pay for what you don't need. Sure, it makes it more expensive for the people who do eventually need service, but that's no concern of yours.
We don't have governments because they're required by God or Nature or anything else. We have governments to simplify an otherwise nasty problem of cardinality. It's just easier to have a government, in most cases. In the computer industry, we hear a lot about "peer to peer" systems. You may not realize this, but the more "distributed" a system is, the less efficient it is (at least when we're talking about loosely distributed, adaptive networks like those used for file-sharing.) In the early days of piracy, we had servers: bulletin-board systems (BBS) were some of the first, followed by internet websites with FTP servers overflowing with pirated software, and later Napster, which held a central repository of available files (an indexing service, like a library's.) If you only have to look for something in one place, the problem of finding something is simple. If you have to look in multiple places, and even have to find out (each time) where to look, it gets expensive.
Governments serve as the central repository of lots of information, decision-making, and services. When you're running water pipes in a city because we've decided everyone should have water available at their house, it's simply easier to have a single agency handling the whole thing. Sure, you can (as a lot of places do) fake their being several different providers of the service, but they often do little more than have their own billing and technical support staff, while sharing (and leasing) the same central resources from city, county, state, or federal governments. It would be costly and dangerous to have anyone and everyone providing the service of running water pipes to your house -- who knows what might happen to the shared resource (damage to the line, leaks, poison, etc.)
What's my problem with the shared-building thing, then? It's that we have socialism disguised as capitalism. I grew up in europe; I'm quite accustomed to the idea of a government owning and running businesses for profit (at least in theory -- it's often at a loss) -- in the US, we have Amtrak as an example of a "corporation" run by the government. My problem is with the "run government like a business" idea, which is that businesses bring in money, and having the government engage in profitable trade would reduce the tax burden on the rest of us, therefore the government should supply itself with income through profitable trade on the side (that is, other than exactly the services it's already paid to provide.)
When a government decides to own a business, it is generally competing with other businesses. What profits it brings in are profits not being brought in by other companies or individuals, who would be paying taxes. A government's fundamental mission, however, is not profit: it is service. An ideal government would make no profit, take no loss, and cost exactly as much as is necessary to provide the services it is expected to provide (the list of which is another matter.) A government is just a public entity expected to receive funds and use them wisely for the public good, not its own. It is not expected to invent its own goals, nor to ignore the needs or wishes of the community. Businesses frequently re-invent themselves when they're no longer profitable, they branch out with new services of their own volition, they attempt to maximize profits by reducing costs while not reducing prices. Their motives are their own, not the community's.
On the positive side, we can consider the following. If a government is required to provide services which are already provided by one or more companies, such companies might be hired to provide those services. The companies being contracted would naturally want a profit, that is, a cost above and beyond the cost of materials and labor directly involved in rendering the services. A government-owned entity would not have such a requirement, and could provide the same services for cheaper. The public benefits from this in the form of lower taxes. Also, a government, at least when "by the people, for the people", may alter its practices to meet public demand, based on voted-for requirements. Capitalist ventures are not required to do so, and might not even have an incentive to do so (some public demands may not be financially advantageous to a company, or the risk involved in changing anything may seem too great; there is no guarantee that someone else will pick up on the gap and fill it in with a new company, only a vague hope.) A government is therefore better suited to making decisions to actively meet public demand, whereas businesses passively meet the demand by trial-and-error, and a lot of luck. You can petition your government and expect some useful reaction; you can petition a private business and expect it to continue to do as it pleases, whether or not that happens to be what you wanted.
There are different levels on which a government operates, and they should perhaps be considered separately. There are services which must be, within a given realm (state or federal boundaries,) rendered by exactly one entity. In those cases, the governing entity for the area sets up a system to provide that service. Unless it's a common "type" of service, there will likely only be one entity like it. It may, on the other hand, be a type of service which private companies could offer, as a slight modification (or specialization) of their existing business. One would need to be selected as the "official" provider of such services. Then there are services which are cheaper when provided by a central authority, such as water, gas, electricity, or other "public works." There are also services which we consider absolutely essential, even though they could be provided in the private sector, and which must be guaranteed to be available. In some countries, this includes health care: although private hospitals may exist, the government is required by its citizens to provide ample health services coverage, regardless of whether or not the private sector wishes to do so as well. Thus, the government may own quite a few hospitals which are run at low or no cost for medical necessities, while the private sector may "fill in the gap" for non-essential care, or for patients who simply don't like the services provided by the government.
[I now realize why this remained 'temporary' for so long -- I never decided where I was going with it. It might be helpful to note that 'libertarians' would prefer a world in which the government practically didn't exist and everything was privately owned, down to the last tree. Other parties vary on how much they believe should be privatized; you don't often hear of privatizing the army, nor of nationalizing all food production. But it's a fairly smooth spectrum, not a radical change in ideology.]