The conjugation of a verb often marks person and number. Is there a language where one can actually separate these two traits or are they purely analytical? (e.g. one could image a language where a verb would be of the form Stem + person marker + number marker)
What you're describing is the very essence of the difference between inflectional and agglutinating language. Thus there are many languages that separate person from number, gender, tense, etc. The person/number syncretism distinction alone is captured in WALS: http://wals.info/feature/29A#2/16.6/148.5. In Europe, Turkish, Finnish, Hungarian and Basque all have this feature. But it is found in languages all over the world.
It's one of the foundational dichotomies in linguistic typology although things have moved on a great deal in that field since then, extending far beyond a relatively trivial point of morphology.
However, note, that not all agglutinating languages necessarily have separate morphemes for person and number. Swahili is a broadly agglutinating language but has person/number syncretism. Just another example that there are no pure morphological types among languages.