The shliach tzibur (shatz) davens 2 shemona esrei's.
1-Which is the one that he davens for himself personally?
2-Are both of the 'real' so to speak? If so, how does this work?
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The Radvaz says (in relation to a different question) that the first silent one is the one for his personal obligation, under the principle of תדיר, and the second is to fulfill the obligation of those who couldn't daven to themselves. It doesn't seem plausible that he would say that in only the case he was dealing with (where the Shatz missed the previous tefilla and needs to make it up) but rather that this is the regular case.