Do you use them?
What sort of weight do they carry with regards to overall content, site structure and SEO?
Thanks in advance for any help.

Title elements provide the information about the link when you hover your mouse over hyperlink.

I do use title element for my users only, but they don't carry any extra weight.

Here's explaining why!
http://www.searchenginejournal.com/how-to-use-link-title-attribute-correctly/7687/

HI.
I do not use them.

Title elements provide the information about the link when you hover your mouse over hyperlink.

I do use title element for my users only, but they don't carry any extra weight.

I'm still using them

We just use title element in hyperlinks for user only. yes, They don't carry any extra weight.

While creating hyper link, make sure that your link is working properly, also, use multiple keywords for be natural linking.