Hello,
No I would never send a pen to a celebrity. I don't feel there is any need to do so. The celebrity will have their own pens, plus I have never had any problems with not being able to make out the signature. Maybe some of the pictures you are referring to were just too dark for the ink to show properly, so my advice would be not to send a picture with a dark background

It's up to you if you want to send your own pen, but like others have said it would cost more, and you probably won't get it returned. I would personally feel really cheeky asking the celebrity to use my own pen, like a diva demanding it because their pen isn't good enough

But that's just me! So no I would never send my own pen, but it's up to you if you want to!