If you're defining people as being something (in this case cugini), no article is needed in the plural. Just like in English, though, you do need an article for the singular form.
Siamo cugini - We are cousins
Sono un/il cugino - I am a/the cousin
One exception I can think of is with a more detailed definition. Ad esempio:
Siamo i cugini di Alessandro - We are Alessandro's cousins
Am I right in thinking that you can replace the last vowel to differenciate between gender and amount?
One male cousin = cugino
One female cousin = cugina
Several female cousins = cugine
Several male (or a mixed group) cousins = cugini
Is the "Noi" necessary in this case, or would "Siamo cugini" be just as acceptable?