How precise and how accurate are typical altimeters these days? That is, if two planes are flying (side-by-side, very near each other), and both altimeters show the same Flight Level, what is the maximum vertical difference in actual altitude you would expect to see between them? (I.e. is it on the order of a few meters, a few tens of meters, etc?) Is there any legal standard for precision/accuracy of the altimeter that must be met (let's say in the US)?
While Accuracy and Precision are closely related, they're not the same thing.
In the context of your question both are important:
In the US altimeters are broadly required to comply with FAR 43 Appendix E limits for accuracy (technically this is regulatory for IFR flight, but TSO'd "sensitive altimeters" will generally meet these requirements - at least when they left the factory - and that's what you're going to find in most aircraft).
The accuracy requirements provided in FAR 43 are based on mechanical altimeters, and are subject to a number of requirements (friction error limits, hysterisis limits, etc.), but the test tables there are a good guide to the accuracy of "typical" altimeters.
Also note that the accuracy of sensitive altimeters depends on the pressure setting the pilot enters (an incorrect pressure setting could mean substantial error). In "the flight levels" everyone uses the same standard altimeter setting (29.92 inches of mercury / 1013 millibars) to avoid this issue (similarly the ATC transponder transmits an altitude assuming a pressure of 29.92 inches of mercury - ATC's computers correct that value based on local altimeter settings as required).
Depending on the design digital altimeters may also be immune to some of the accuracy-affecting issues in an analog/mechanical altimeter - for example a solid-state digital altimeter should not suffer from "friction error" as there's no rotating/moving shaft involved.
Analog altimeters are "reasonably precise" -- take this example:
The tick marks are every 20 feet. One could easily eyeball 20 foot increments, and even 10 foot increments (with the pointer directly between two ticks). Similarly the pressure settings are marked every 0.02 inches of mercury (or 1mb), allowing for a fairly precise setting of the value.
Digital altimeters are extremely precise (often offering one-foot precision in their display, and 0.01 inches of mercury for the pressure setting).
I believe the precision requirements for altimeter markings are specified in the relevant TSOs, but I don't have those documents handy.
In both cases the precision of the markings can exceed the accuracy of the device: At 10,000 feet the FARs allow an 80 foot tolerance, and whether your measurement is presented as a needle position (roughly to the nearest 10 feet) or a number (to the nearest foot) you could still be 79 feet higher or lower than you "think" you are and be "within tolerance".
So, combining all of this together - If we have two aircraft flying along indicating the same altitude (say 10,000 feet) and both altimeters are perfect there will be zero feet of vertical separation between them (the static ports of each aircraft will be at the exact same altitude).
On the other hand if both aircraft are at the outer edge of tolerance (one +80ft, one -80ft) there will be 160 feet of vertical separation between them (discounting visual "slop" in the analog altimeter and assuming the pointer is perfectly on the line and the pressure setting is perfectly dialed in).
(In either case the answer would be "WAY TOO CLOSE!")
Well, it depends a little bit on how you define accuracy. Working altimeters are as accurate as the pilot makes them. In perfect use, they work great, in typical use, they work a little less great, but still pretty well.
First of all, the altimeter is part of the pitot-static system, so that means they are dependent on having smooth airflow, and the pitot-static system being working (not clogged).
Second, altimeters in GA airplanes, not sure about others, require you to manually set the pressure of the atmosphere (barometric pressure), to get the correct reading. This can be done with some accuracy, but usually never 100%.
Finally, altimeters in GA airplanes are required to be read in a similar manner to an analog clock. So there is a little pilot error to consider.