[KyleGoetz]

I swear this exact same thread has been made within the past six months...weird.

If you are asked これは〜さんのペンですか you can answer
いいえ、違います。
いいえ、そうではありません。

You don't need to use those 私s, これs, and それs in your answer—they make it sound non-native.

ではありません means "is not"
違う means "is different"

Even in English, both work:
Is this your pen?
No, it's not.
No, that one is [a] different [one than my pen]..

Personally, my go-to answer to this question would be 違います because it's the fastest to say. When talking with my friends, I probably equally use いいえ、そうじゃないっす and 違う.