For WiFi MIMO AP's, the 2 transmit RF streams should be independent. I.e., they are trying to transmit different info, with instantaneously different amplitides and phases (but on the same frequency). If this is like LTE MIMO (and I think it is), then the (receiving) client device will report back on how well it can demodulate both streams; if that is not working, then that link reverts to SISO with just one data stream....preferrably transmitted from one antenna so that there will be no odd phasing/pattern effects between the 2 antennas for that data stream that will cause pattern distortion.However, other client devices working on the same AP may have physical links good enough to utilize MIMO and so their 2 transmit data streams will still be on both antennas. (And I am going on some very basic knowlege of 802.11, and assuming there is no tranmsit time multiplexing of the different client device's data steams.)
What this all boils down to is that the RF from each antenna SHOULD (as I understand it) be fairly statisitically independent in instantaneous phase and amplitude. As such, the phasing of 1 signal is random compared to the the other. So the 2 antennas should not statically phase in or out with each other and produce any sort of steady figure 8 or cardiod or any other distorted, non-omni pattern. They might phase oddly for a symbol period now and then, and produce an odd pattern for an instant, but that will look like some sort of multipath fading to the receiving client device.
As for VSWR, yes, I can see one effecting the other. The matching to each antenna might have to be modified for that. And I guess there COULD be pattern distortion caused by the other antenna acting as a passive re-radiator (like in a yagi) but that would vary with how the 2nd antenna appeared 'terminated' for energy that coupled from the other antenna; I am not sure how to think of that off the top of my head. Also, the fact that these are broadband signals means that any such passive re-radiator pattern effects would probably be localized in spots in the transmitter spectrum and would not be an issue for the whole transmitted spectrum. So, my general thought (with no data!) is that is not a big issue either.
So that takes care of the pattern distortion issues; it should not be huge, as I understand things. And the VSWR can be corrected. BUT....There is the question of the transmit energy from one amplifier getting back into the other via the antenna coupling and causing some havoc between the amplifiers. (And I assume this is a modern AP which can transmit 100 mW or more on each channel, so a real amp is needed on each channel.) The energy feedback from one to the other can cause some real problems; the most common is that there will some undesired mixing of signal 1 with signal 2 at the output of signal 2's amplifier, and vice versa. Since these are on the same frequency, there should not be much of what would be considered intermod, like we commonly deal with in 2-way systems between two different transmitted frequencies.
BUT there will be what is called spectral re-growth, where the adjacent channel, and 2nd adjacent channel, and out to the xth adjacent channel's, transmitter noise levels will go up above what one of the 2 channels would transmit alone; this can get really bad, depending on the level of coupling and the amplifiers' behavior. This IS a problem, both for FCC compliance and for system performance. In that case, you are basically polluting other channels in the WiFi spectrum with sideband crud caused this RF 'crossover' feedback between amps that is being coupled back and forth via the 2 close antennas. So you need to be able to measure this as a designer, and be ready employ methods and components to limit the transmitter to transmitter coupling via the antennas (or any other internal AP path for that matter).
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