Learn Polish: My bardzo się naśmialiśmy wczoraj wieczorem.

The base verb is śmiać się (to laugh). By using the na- ... się pattern to form naśmiać się, the verb takes on the meaning of doing the action to the point of satisfaction or excess. Instead of just 'we laughed', it means 'we laughed a lot', 'we had a good laugh', or 'we laughed our fill'.

Why is się placed before the verb rather than after naśmialiśmy?

The reflexive pronoun się cannot be the first word in a sentence, and it generally avoids being the very last word if possible. It naturally gravitates toward the second position in a clause. Tucking it between bardzo and naśmialiśmy flows much better to a Polish ear than putting it after the verb.

Since naśmialiśmy already contains the 'we' ending, why is the pronoun My used at the start?

You are right that the -śmy suffix completely identifies the subject as 'we', which is why Polish usually drops subject pronouns. Including My here is done for emphasis, making the sentence sound a bit like 'As for us, we laughed a lot'.

Could you use naśmiałyśmy in this sentence instead?