Am I correct in thinking that the lower frequency sound that makes it to either channel and is processed as "side" would be more reverberant? So while there's less bass present in the side component, the bass that is present is muddier.
Yes, the Side channel will contain more reverberant content. Not just at low frequencies but across all frequencies. The Mid channel has reverberant content as well, yet has a higher proportion of direct sound so it will usually sound less reverberant. And yes there will usually be less very low frequency content in the Side channel
I like approaching each aspect separately, using a tool like MSED in "tru" mode for overall stereo width adjustment, and using a Mid/Side EQ for adjusting the timbre of the direct center-image sound versus the more ambient wide sound.
But a Mid/Side EQ can also be used to adjust width (and to some degree reverberance) by frequency range without changing the overall tone by applying
inverse (mirror image) EQ curves to Mid and Side, boosting where the other cuts and vice-versa. The EQ plugins Noah points to can be used to do that, although you'll probably need to dial in both the original curve on one channel and its inverse on the other yourself. No unlikely that you might want to adjust both width and tone, but this approach allows you to separate the two.
Does anyone know of a Mid/Side EQ that has an "inverse curve" mode which automatically creates an inverse curve in the second channel? That would be very useful for this as one could adjust width by frequency with gain being automatically compensated, separating the width/rebererance adjustments from the tonal adjustments.