时量补语：modal and verb 了
In practice, there is no difference. Both sentences mean

we have been separated for 10 years

and that's the information most natives would get from it.
In theory though, textbooks and/or particularly pedantic speakers might point out that there is in fact a difference: the presence of the first 了 as a marker of aspect could signal that the separation has ended. Thus, you might get two different translations:

我们已经分别了十年了
We were separated for 10 years (implying now we're back together)

and

我们已经分别十年了
We have been separated for 10 years (implying we still are)

However it can be argued that the presence of the adverb 已经 (already) helps discriminate in favor of the latter, i.e. that being separated is an ongoing situation.
Anyway, as an unrequested comment: personally I would use the verb 分开 here, instead of 分别