I love this idea. It’s the dream of many and the reality of a growing group. Some have mentioned that Micro.blog does this out of the box, but I’ll mention that I have some tools that allow me to do it outside of that. I use WordPress for my website, but it dovetails well with social readers like Aperture, Indigenous, and Together. Small standards and building blocks like Microformats, Webmention, Micropub, and Microsub glue it all together. Here’s an overview of what some of it looks like: A Twitter of Our Own (short video) along with slides. Those with some technical expertise should be able to get this up and running for themselves. If it’s your dream, I hope you look into the solutions and come join the growing community.