This idea/suggestion has been flagged as Implemented so it has (or will be) implemented into the game in some capacity. More information can be found in the post from the development team.
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Proposal
Container are currently not accessable by other characters if there is a "NoTrade Shared" item im it. But you can drop those items on the floor or trade them directly to characters on your account.
Change containers so that they are usable by all characters of the same account
Justification
QoL improvement as it's already popssible to trade the items.
Motivation
I hate 20 Harvester Deeds on the floor.
Nobrainer, everyone has to upvote!
 
The Development Team has provided a final answer on this suggestion. Votes are no longer accepted.
Go one step further, have the bank be Shared. Star Trek Online has always had a shared bank, although it was smaller.
 
So these are actually two different things:
1) Containers are currently not accesible if a "No Trade" item is in it - except for the owner of the house. This is intentional
2) No trade shared items could only be picked up from the floor. This was half-intentional to avoid messing around with NTS items.

We have relaxed 2) in the upcoming patch, but reserve the right to reverse that decision if we find any malivious behaviour around it
 
Could we still let players open containers with no-trade items in them, but only take items they own or are allowed to move?

Example: Player 1 puts a no-trade item in a container that also has Player 2's tradable items. Right now, Player 2 can't access the container at all.

Maybe keep the container accessible through normal housing permissions, but darken or lock the restricted item in the inventory UI if there is a non-transferable item.

(early morning brainstorming, i think there is a flaw in my plan lol)
 
Could we still let players open containers with no-trade items in them, but only take items they own or are allowed to move?

Example: Player 1 puts a no-trade item in a container that also has Player 2's tradable items. Right now, Player 2 can't access the container at all.

Maybe keep the container accessible through normal housing permissions, but darken or lock the restricted item in the inventory UI if there is a non-transferable item.

(early morning brainstorming, i think there is a flaw in my plan lol)
No