Is the sentence right or wrong?
Children, be careful, don't hurt yourselfs!
Singular pronouns end with -self, plural end with -selves.
There are two people and the action has this direction: ↔︎ We do not want to repeat the verb twice, so we use "each other".
"Each other" doesn't change, it is always in the same form.
I help you and you help me. → We help each other.
She knows him well and he knows her well. → They know each other well.
Julia loves Frederic and Frederic loves Julia. → They love each other.
Bill and Steven have had an argument. Bill doesn't speak to Steven and Steven doesn't speak to Bill. → They don't speak to each other.
My husband trusts me and I trust him. → My husband and I trust each other.