Does the Earth receive a net charge from the aurorae?
No. If that were the case, the Sun would eventually develop a net charge. Let's imagine that's the case and the Sun develops a net  positive charge. That charge would eventually stop negative particles from flying away and facilitate the ejection of positive ones. This regime would then restore the neutral charge of the Sun.
This negative feedback makes sure the Sun remains neutral and that Solar winds carry no net charge.

See this Q&amp;A for more details:Does the solar wind correspond to a net electrical current?