It means what you see is what you get, she doesn't have a hidden personality, or is not two faced, saying one thing to you and another behind your back.
Although I'm not sure this applies to Elsie, you should hear what she says about you!
The usual meaning of 'side' is pretentiousness or arrogance. That is, if someone 'has no side to her', it suggests that - not even if she might be entitled to do so, as it were - has a woman any inclination to show she is superior to others. So, a professor might treat a teaching assistant as an equal rather than as an inferior.