I have access to my own hosting equipment; I have a home server for private / low-bandwidth stuff, and I have a server at work which gives me all the hosting goodness I need. I've already experimented with sending data from my arduino project to my server and displaying it within a simple html / php page.
What I'd prefer though, is to just keep using twitter. It does the trick, it's convenient, and I don't have to reinvent the wheel.
Personally, I like GizmoJoe's idea. If Neocat's setup (the server side) was open source, I would try and set up a copy of it on my own server. That way I wouldn't be using Neocat's bandwidth. (I don't mind using Twitter's bandwidth, they monetize my tweets with ads and whatever else they stuff into my stream.)
As I mentioned earlier, there was a period of very inconsistent behavior a couple months ago that got me looking into other options. I made a note back then about a linux based command-line twitter client. I'm pretty sure I could utilize that by just reconfiguring how the Arduino sends its data, and setting something up on one of my servers to accept the data and pipe it into the command-line client.
When Neocat's system started working ok again, I stopped looking into the alternatives but it might be worth persuing that again now.
Cheers!