It depends on how you travel to the Ethereal Plane.
Nothing about the Ethereal Plane's description indicates that objects are forced to leave, but how you get there matters.
As I see it, two categories of planar travel are relevant to answer your question.
This includes Portals, spells like Plane Shift, and traits like a Phase Spider's Ethereal Jaunt.
In all these cases, magic brings you to the Ethereal Plane and then no longer affects you.
Someone with Detect Magic in the Ethereal Plane cannot sense any ongoing magic on you related to the planar travel, because there is none.
In these cases, you and anything you place in the Ethereal Plane are there permanently and can't leave unless another magical effect lets you.
This includes spells such as Blink and Etherealness.
In these cases, magic brings you to the Ethereal Plane, and keeps you there for a duration.
Someone with Detect Magic in the Ethereal Plane can sense the ongoing magic that keeps you in the plane.
When the magic ends, you and everything you brought with you return to whatever plane you were in before entering the Ethereal Plane.
In particular, both Blink and Etherealness have a range of self. The objects you carry are not the target of the spell, they just travel with you because you are carrying them. Both these spells cannot target independent objects, so if you are no longer carrying an object then it is no longer affected by these spells and it will return to whatever plane it was in before entering the Ethereal Plane.