I manage 4 Twitter accounts - 1 personal, and 3 professional for my company as a Social Media Specialist.
I see NO harm in getting a Twitter account. Personally, I believe it's less invasive than other forms of Social Media because it's primarily used by business people and marketers.
Yes, there are people out there carrying on personal business. But, the great thing about Twitter is that you can log-on anytime to see anyone's profile except for Direct Messages. For instance, Chris Brogan is a thought leader in the world of Social Media. here's his twitter address: http://twitter.com/chrisbrogan
You can see all of his followers, all the people he follows as it's all public knowledge.
I'd not have a problem if I had a 13 year old. But, I'd want to find out why they want one. FB is MUCH more social, Twitter is MUCH more business-related.