俺、私 being used by the other genders

I’m aware Japanese pronouns are not strictly gender specific but I don’t understand how males using 私 and females using 俺 changes the meaning

私 is used by males in formal settings, I read spmewhere. Is there more to it?

I’m mostly confused about 俺. Does it give the context some harshness or something similar, since 俺 is informal? If so, is the reverse also true for 私？