ELI12: how do libertarians want wages to work?

Just to get it out of the way: groups don't have ideas or opinions.  Individuals do, and some self-identified groups share many, but rarely all and rarely levels as the identified individuals do.  This is true even for traditional groups like "liberals", but even more so for disorganized ones like "libertarians" or "rationalists".  You can find the US Libertarian Party platform at https://www.lp.org/platform/, but the vast majority of people who admit to libertarian leaning that I know of explicitly separate the libertarian belief cluster from the Libertarian party. 

The general idea (with many variations) is that it mostly works as today, just with far more individual freedom to have exceptions, and less government enforcement of specific equilibria.  Unions may still exist, but they're voluntary organizations to help employees coordinate their negotiations.  The minimum wage isn't enforced, but it probably still exists as a norm for "the lowest you should expect to pay a competent adult".  

The ultimate private property is yourself, and it's offensive to be told that you can't charge between $0 and some arbitrary number for your time.  ESPECIALLY if the ways you can spend your time (due to preference, availability, or ability) are worth just under that, in which case you're being told that you can't work at all.

You don't mention which libertarian works you consulted in forming your views on the topic. A very accessible introduction is "What it means to be a libertarian" by Charles Murray.

This point of view is very old e.g. the early Daoist works have libertarian threads. So you don't have to imagine what libertarians think, and they have been thinking hard about the issues for a long time. 

One thing that surprises many people is the enthusiastic support among many libertarians for collective action and for cooperative organisations. The caveat being that they are not mandated by overbearing people with guns.

Another thing that often surprises is the realization of how well people are able to coordinate among themselves without too many people who are from the government and here to help.

It seems like unions and minimum wages are a form of bargaining, using the government as a way to overcome coordination problems given the cognitive costs of doing distributed game theory. Why isn't that considered by libertarians to be a legitimate (i.e., reasonably endorsed as part of a globally coherent notion of justice) form of bargaining / decision-making?

As for myself, the question is not "Is this a legitimate coordination mechanism?" but rather "Does this coordination mechanism effectively solve coordination problems?" In other words, rather than viewing government as some sort of privileged system that solves all the problems in the private market, we can view it from the same perspective that we view systems in the private market. We can do this through the lens of game theory: model the government as being composed of a bunch of different economic agents who each try to achieve their own goals in light of the incentive structures in government, and then ask whether the aggregate behavior of these agents adds up to something we prefer to alternatives.

When we start getting specific about what we mean by governance, then the flaws in using government as a device for collective action become more apparent. The flaws in first past the post representative democracy are fairly well known, but I'll repeat some arguments here:

1. Given typical election margins, the expected selfish value of voting is very small. Therefore, voters have very little incentive to be informed about which candidates might benefit them.
2. Candidate media coverage is a very powerful lever for getting politicians elected. Therefore, rich special interests can sway elections to their desired ends by paying for targeted media coverage.
3. If a candidate wins by even a single vote, they win 100% of the effective power of the office for their term. Therefore, minority interests are given almost no weight during democratic deliberations, unless those interests could plausibly lead to the incumbent losing a future election (for example, by people in the majority being sympathetic to minority interests).
4. First past the post voting doesn't take into account the intensity of preferences. For example, the weak interests of 9 voters who are OK with locking up marijuana smokers outweigh the strong interest of the 1 voter who doesn't want to be locked up themselves.

Given these flaws in the way democracy actually functions (as opposed to the way people poetically imagine democracy), there's no guarantee that coordinating through it will actually lead to outcomes that maximize utility, like happiness or preference satisfaction. By contrast, libertarians generally share the intuition that market structures usually do a better job at solving coordination problems than government (and sometimes back it up with results like the fundamental theorems of welfare economics).

Not at all related to any libertarian or other political view point.

There's a huge literature for labor economics that will have a lot of theory on wages and employee-employer market workings.

Some of the more simple approaches is to assume wages and labor services are no different from any other price and service/good setting.  You can add another layer here related to reservation wage which then puts the floor in. One might argue that some market failure situation exists that makes the reservation wage fail to bind market wages so government imposed minimum wage should be set to the appropriate reservation wage level. (Note, I am not aware of anyone actually making that argument but think one would find bits or even all of that in much of the literature.)

I think perhaps a more interesting line might be something of a mixed model where one analyzes the internal compensation/pay policies for the employer as a distributive problem rather than the allocative problem that often seems to underlie wage models. Then across the industry/sector you would see something of a market wage observed but the formation would be driven by something a bit different than the standard econ 101 type story. I'm also not aware of any model like this but would expect that at least some of the labor econ literature at least implicitly includes it.

I'm amazed you only have 4 answers so far.

The bog-standard classical (and, yes, "libertarian") answer is that wages work exactly the same as all other prices - prices for candy bars, gasoline, houses, lawn mowing services, plumbers, and milk.

That's to say, supply and demand (sellers and buyers) set prices, same as with everything else. If buyers don't like the price, they shop around some more, settle for a lower-quality "product" that's cheaper, or offer more to get what they want. If sellers don't like what's on offer, they look for another buyer who'll pay more, try to improve the "product", or settle for what buyers are willing to pay.

Generally speaking, price controls (a minimum wage is a price floor on labor) make things worse for pretty much everybody - laws that make mutually-advantageous trades illegal are generally a bad idea. Plenty of countries don't have any minimum wage (last I checked, those included Denmark, Iceland, Norway, and Sweden - all pretty decent places to live).

There's a whole academic field dedicated to studying and understanding this called "economics".

The basic idea is that without gov forcing out competition via monopoly the market provides arbitration services.

There's a few countries without minimum wage laws that have way better average wages than the US and many others, I believe this fact already invalidates some of your observations. 

For libertarians workers unions would be ok, as long as they're not government sponsored (they're a type of private government), while government intervention is pretty much always seen as bad since they are not considered a legitimate way to solve market issues (wages are a market issue)