It depends. Usually, people doesn't mind sharing their source code for Processing, but if you want to keep it private, that's your right.
.exe can only be run on Windows, so it is not good.
Applet form is better if your application doesn't do I/O (reading/writing files).

There are some sites to share sketches, but usually it means you must disclose the source, and sometime to use their license (GPL for OpenProcessing, for example).

It depends how hard you want to do that...
If you have some money and some time, you can just buy the right to use some domain name, eg. Snallow-Processing.org or something, then find a host, build a Web site there (a simple WordPress installation can do the trick, most hosts can install it for you) and show your applet. It isn't that expensive, but it can take some money and need some knowledge.
On a more modest level, you can find a blog host, like WordPress.com, but you have to be careful, on some it can be hard to install the jars and put the right code to embed the applet. There are some threads on the topic in the forums (old and new).