In my game, in which I play a wizard, I have met an NPC wizard who is willing to let me learn Leomund's Secret Chest from her spellbook. This got me wondering what would happen if I learned the spell and cast it on the same large chest that she had already cast it on (using my own separate tiny replica). Would this enable us to effectively teleport objects back and forth by alternately summoning the chest and placing things in it, then letting the other person summon it and remove the items? Or would something else happen? How would the chest work if 2 different wizards cast Secret Chest on it?
The most "potent" effect prevails
Under the rules for Combining Magical Effects, two effects with the same name don't stack. QED.
The only tricky part is the DM has to decide which is the most "potent". If one caster used a higher-level spell slot, that would be the most potent.; otherwise, the "most recent effect applies".
In principle, this would allow both of them to summon the chest.
Leomund's secret chest doesn't restrict the chest from being used as the component for multiple castings of the spell. You'd just need to summon it back so the other wizard can cast the spell.
The ability to summon the chest only works "while the chest is on the Ethereal Plane", so if you speak the password and the chest doesn't appear, probably the other guy is using it.