basically they give you the relevant recruitment ability no matter what faction the government building belongs to even when they are damaged, but you dont get the plus or minus bonuses to law/trade etc.
so you need to destroy them to prevent being able to exploit the recruitment.
personally i repair the government buildings when they are the same faction as mine, eg. if playing as the aedui you take a rebel town with an aedui government complex. this is officially considered cheating though i believe. playing as koiine.... err... the greeks, i found a few hellenic local tyrant complexes in some towns i captured as well.
however keep in mind the AI gets to take advantage of this with pretty much every town it takes.
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