One of our clients asked me if I thought it was effective to reach out to new Twitter followers with a Private Message. I wasn't sure at the time but I've been curious as well. My personal experience has been that the majority of the PMs I receive are automated and impersonal. To me Twitter is such a short lived experience, I usually just reach out to followers publicly to thank them for following.
My inbox is mostly filled with offers and sales pitches. The better PMs say hi, link me to a website or another social network. These seem to be the most friendly. The worst and more common want to sell me eleventy-trillion followers - ugh! Twitter discourages spammy, automated messages and abusers can have their accounts suspended.
The first PM is selling me a very specific type of insurance, and the second wants a retweet. The third is nice but clearly automated. The fourth is selling me something, and the final request is to like a Facebook page.
From where I sit, private messaging in Twitter is so filled with automated messages, it's just not worth my time to check it.
UPDATE: Just to be clear, I still check it! Never want to miss out on something, but I do sure get a lot of offers for 30K twitter followers for $10. ;)