It depends on whether you're using the chromium-daily PPA or you used Google's deb (the former looks for Firefox's libflashplayer.so, the latter needs you to install it).
If you're using the chromium-daily PPA (https://launchpad.net/~chromium-daily/+archive/ppa), Chromium will look in a bunch of folders[1] for plugins, including ~/.mozilla/plugins. I suggest you get the tarball from http://get.adobe.com/flashplayer/ and put its libflashplayer.so in ~/.mozilla/plugins. Chromium will look here for it upon startup and you're set.
If you installed using Google's deb (http://dev.chromium.org/getting-involved/dev-channel), it doesn't yet search the Mozilla plugin dirs, and thus you need to make an /opt/google/chrome/plugins dir and put the libflashplayer.so in there and it'll work.
Edit: if you're using Google's deb, you have to run google-chrome with the --enable-plugins switch. The deb adds their repo to sources.list.d, but I have no idea when they'll enable plugins by default in their repo. I'd ride the chromium-daily channel, since lots of shit is broken anyway.
For those running Ubuntu AMD64, you can put both 32 and 64-bit versions of libflashplayer.so in ~/.mozilla/plugins (call them libflashplayer32.so and libflashplayer64.so if you like), and 32-bit Firefox will load the 32-bit plugin, and Chromium 64-bit will load the 64-bit plugin. You can get the 64-bit Flash plugin here:
And if anyone's wondering why I'm not running 64-bit Firefox 3.5, it's because the Ubuntu (and possibly Debian, I've no idea) packagers still have it branded 'Shiretoko', which throws off every site searching for 'Firefox' in your UA. Hopefully by Karmic's release they'll have this sorted out.
I have --enable-plugins turned on atm and I don't see any difference between chromium's flash and firefox's, except that when the plugin crashes on chromium, I reload the page instead of the entire browser.
