Before I post something on the web, I always convert it to sRGB. Otherwise, your image will not look right. Of course, this is just to please the masses. If I were posting photos just for myself, I'd convert to my own monitor profile, instead.
Incidentally, this problem isn't just a web problem. Certain image viewers are profile-aware. For example, ACDSee on the PC is great if you are looking at sRGB images, and if you are using a relatively uncalibrated monitor that happens to match sRGB. However, on my setup, everything looks wrong. If I use something like Photoshop of BreezeBrowser, which can be configured to "know" about your monitor profile, and is aware of the color profile of your source image, it will do the conversion to the monitor's color space.
Depends on the app. And probably the platform. I'm not sure what happens on the Mac.