no, it is not necessary. You can say :

Mon ami et moi sommes allés à...

Note that if you use the "nous", it becomes an emphasis and you should use a comma :

Mon ami et moi, nous sommes allés à ... 

You can say both but be aware that the first person plural has mostly disappeared from conversational French so the most usual way to say it would be:

Mon ami et moi, on est allés…

In that case, the pronoun is mandatory as you can’t say "mon ami et moi est allés…"

The only case where you would have to repeat the "nous" would be for a self-reflecting verb. For instance, "nous nous sommes habillés" ("we dressed ourselves", or more correctly "we dressed") would transpose into "mon ami et moi nous sommes habillés" ("my friend and me dressed ourselves", or more correctly "my friend and myself dressed").