A workaround has been created, but it is not pretty. To bypass the
protection, you need to download the extension or userscript to the
local system. Once there, you need to drag and drop it into the
chrome://chrome/extensions/ page in the browser. If you drag and drop
it elsewhere, nothing will happen.
When you drop it on the extensions page, you will notice an install
option popping up there.
When you select that option, you will see the standard installation
dialog that informs you about the rights that the script or extension
Sourced from: Chrome Fix: Extensions, apps, and user scripts cannot be installed from this web site.
Edited on 20121125:
I came across another way which requires starting the browser from the command line.
For example, if you are using Chrome and want to install an extension not from the Web Store, the following code entered in a terminal starts Chrome in a "permissive" mode for this instance:
Explanation for the switch:
Enables extensions to be easily installed from sites other than the web store. Without this
flag, they can still be installed, but must be manually dragged onto chrome://extensions/.
The usual caveats apply. The user should be careful about installing extensions. Commmand line switches come and go and need not work with earlier versions or future versions of the browser.
I can't answer about Chromium because I don't use it.