Don't know about 1Password, but LastPass encrypts all passwords client-side. Only marginally less safe than storing passwords in an encrypted file on your personal machine.
<BTW: This goes without saying, but I'll say it anyway: Don't lose your 1 master password or your password database. ;) >
At first I worried about what would happen should the db file get corrupted somehow. But then I set up the backup tools with which I can keep as many backups in as many places as I want, so unless the db file gets corrupted in all places, I think I will be fine.
I'll have to give it another look, I guess.