You can easily say мы there, no real problem. Сами is there for style, you know, it's best not to repeat the same word in the same sentence. In colloquial speech people also say things like я сам..., он сам..., она сама..., мы сами... quite often. It's a common construction. You can use мы сами in the но clause of the sentence for that matter.

Я сам! - pretty much means I'll do it myself, like without anyone helping. It's also a common name for kids books that teach how to do simple things.

You can also think that сами in that sentence means "we ourselves".