It is said the same way as in English: "私たち---" / "We ---"
For example, "私たち日本人" is a common way to say "We Japanese". Your inclusion of の was incorrect.
Here are a number of examples:
http://eow.alc.co.jp/search?q=私たち日本人
The same goes for "We Americans," (私たちアメリカ人) and as an added bonus, here is an example with 我
We Americans make no secret of our belief in freedom.
我々アメリカ人は自由への信念を決して秘密にはしない。