It will also get 3 protons (in the case of NH3) or 4 protons (in the case of NH4+). For simplicity, in the case of NH3, N will share 3 of its electrons with 3 hydrogen atoms (each with 1 electron and 1 proton). It doesn't take the hydrogens' electrons from them (at least not entirely--see below). If it did, then you would write ammonia's formula as N3-(H+)3. Covalent bonds involve sharing of electrons. The 3 electrons from the hydrogens serve to complete the nitrogen's octet, but they don't count toward the formal charge. For the formal charge, each covalent bond counts as one electron.
So for NH4+, there are 4 covalent bonds around nitrogen; therefore there are 8 (4x2) valence electrons satisfying the nitrogen's octet, but only 4 valence electrons counting toward nitrogen's formal charge. Add the 2 non-bonding 1s electrons and you have 6 total electrons assigned to nitrogen for the sake of formal charge. Since nitrogen has 7 protons, the formal charge on nitrogen is +1.
(Caveat: there is some polarity in the N-H bond. That is, a given electron is more likely to be found around the nitrogen in an N-H bond than the hydrogen. However, it is far from being a complete charge transfer.)