What does "السلام عليكم أيها النبي" actually mean?

When I first learnt how to pray *salat*, I was taught as part of the *tashahud* to recite the phrase "السلام عليكم أيها النبي." I am hardly fluent in Arabic, but my translation of the above is basically "Peace be upon you, oh prophet." As a native English speaker this has always felt wrong to me, as using the second-person pronoun "you" suggests that I am speaking (and thus praying) directly *to* the prophet, rather than to God Almighty; this brings to me concerns of *shirk*. My question here isn't about whether reciting the *tashahud* itself is required, or exactly what form it should take; I am more concerned about the actual Arabic construction of the sentence itself. Does the use of "كم" (or, as I have also seen used, "ك") in this case have the same connotations in Arabic? Or can it be reasonably used to refer to a third-party in a manner not seen in English?