What's the correct construction?
We sat silently in front of each other, she covering her belly and me covering my pants.
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As so often, it's a question of subject or object.
The possibilities are illustrated:
She doing X and I/my doing Y did not take my fancy. (Subject)
Mum suggested her doing X and me/my doing Y. (Object)
Your example sentence uses the popular me rather than the formal I.
The latter, which is more correct, would read:
We sat silently in front of each other, she (was) covering her belly and I (was) covering my pants.
An easy way to remember if you should use subject pronoun (I/you/he/she/it/etc.) or object pronoun (me/you/him/her/etc.) is to see the location of the verb
[this works for easier sentences, as the sentences become more complex this becomes more difficult to decipher]
Subject pronouns go before a verb and an object pronoun goes after a verb.