I discovered this accidentally, and I don't understand why this is happening:
- when I login locally (i.e. standard tty, no gdm/xdm), I can just hit ENTER, no password, and the system grants access, be it my user or root account (!)
- when I login remotely (FTP, ssh from another machine), the right password is hopefully required, as well with "su" (locally).

I'm running a Slackware 12, 2.6.22.6, Dropline GNOME has been installed, no fancy stuff otherwise.

Any idea why my system shows these symptoms ?

(it would be nice if I could avoid a re-install)

Have you modified your /etc/inittab or /etc/login.defs files?

I'm not familiar with Dropline Gnome so I'm not sure how this might affect logins.

It is quite odd that the root user can log in without a password... even if login.defs is modified to allow no-password logins, the /bin/login program explicitly overrides that for root login. The only way I can see that it could occur is if inittab is using something other than agetty and /bin/login for logging in to the console.

This does, of course, assume the obvious -- that you have actually set a password for root
I'd try explicitly setting a password on the command line after logging in locally to see if the problem persists.

No, except I switched to an init 4 for a while and back to 3.

At least your answer had me do a little more research, and I've found something about a Dropline "*Critical Update* - 2007-12-26" regarding PAM and a "local authentication exploit" ...

I have a root password :-)

Password change has no effect.

I'll see to update my Dropline distribution, and if it's indeed the source of my problems, then I'll accept Patrick's opinion of PAM, forever ...

EDIT: a Dropline/PAM configuration problem, indeed. I missed a critical update ... pam-0.99.9.0-i686-2dl fixes the issue. Otherwise, check manually the /etc/pam.d/system-auth for the line beginning with "auth" and replace "sufficient" with "required" ! (when I thought Dropline had stopped being intrusive !)

That update was issued a ludicrously long time ago, man. You have fun trying to go months without updating your system.