If proteins have an overall charge, how do membrane proteins traverse through the hydrophobic region of the plasma membrane? These two concepts seem almost contradictory, proteins have a net negative charge due to the amino acids in them each having a small negative charge, yet membrane proteins are able to exist traversing the plasma membrane, which is impermeable to charged molecules (hence the need for membrane proteins). How do these two statements remain true simultaneously?

Transmembrane proteins are inserted into the membrane in the ER in a rather complicated system, there is a whole chapter about translocation of proteins in "Molecular Biology of the Cell".

The proteins are moved through an aqueous pore in the Sec61 complex, which explains how charged parts of a protein are moved across the membrane.

The parts of a transmembrane protein that remain in the membrane are usually composed of hydrophobic amino acids that aren't charged.

I strongly recommend you to read up on the whole translocation stuff in a cell biology book like the Alberts I linked. There's a lot more to it than what I wrote here.