Personally, I'd like the weather to be the same for both sides each turn, especially for PBEM. A bad string of weather can be more frustrating than no research/diplo gains.
If this isn't possible (for whatever reason) then maybe it would be possible to weight the changes based on the past turn. For instance, if I encounter mud in Russia on a turn, I would expect it to be very probable for the opponent (either AI or human) to encounter the same conditions on the following turn. If it's clear for my opponent, it souldn't always be mud for me.
I've encountered several games where it is always clear for the AI, and always mud for me starting in September and continuing through into November. This makes progress grind to a halt. Also, mud should be much less common, certainly in the fall at least. You'd think from playing this game that all of Europe and Asia was one huge swamp and quagmire most of the year.