This is basically because plenty of companies are doing this (see https://twitter.com/#!/vodafoneuk for example), so people get used to it.
So really it's not so much posting to a random website, it's more using social media as a means to contact them.
But based on the trend, people have come to expect it as the norm thanks to companies that lead the way in doing it, and now those compares are leading the way in proactively reaching out to customers who tweet not directly at them - for example 9 months ago I tweeted something like "As soon as my contract with Vodafone ends I'm moving to Orange", both companies tweeted at me offering to help - so people could well come to expect that as the norm too.
This cracked me up. And likewise, I think we've seen one. Meanwhile we have nearly a quarter million email "tickets" to date.
Lots of users realize that there is likely a faster response from twitter than [email protected] because the developer has some amount of face at stake with the dirty laundry in public.
If you really wanted to "expose" something in public, you'd put it up on a blog or someplace that's actually on the public facing internet. Not that it would get you any more chance of the company hearing about it, but at least other people might see it.
And, of course, if you run a company that simply doesn't respond to things on Twitter, the customer in question will hopefully learn that they can send you an email and get a fast response.
Can anybody else verify this? To me, it seems ridiculous that anybody would expect to get support by posting something to a random website. Personally, I go to twitter.com about four times a year and type in the names of my products to do a quick vanity search about what people are saying about them. I've never seen anything like a support request (or even a complete coherent thought) in there. It just doesn't seem like something worth monitoring.
My product sites all have a contact page with an email address on it. If you want to contact me, that's how you do it.
Amazon has forums with dedicated representatives monitoring them. That's how you get in touch with them. I've never gone more than a few hours without a response from somebody who knows what they're talking about in there.