It can only be your gear and only you benefit
The description of the primary Rune Carver feature says:
You can use magic runes to enhance your gear
I admit that the wording that comes later merely says you can choose a number of objects equal to the runes you know, without stating it can only be your equipment. But the description of each individual rune makes it clear that only you can use the rune or benefit from it anyway.
For example, the Fire Rune says:
While wearing or carrying an object inscribed with this rune, your
proficiency bonus is doubled...
In addition, when you hit a creature with an attack using a weapon,
you can invoke the rune to summon fiery shackles
There is an argument that the previously mentioned paragraphs can be taken separately, in which case you could put a rune on somebody else's weapon which would prevent you benefitting from the first aspect (the proficiency bonus) but could still activate the rune when you hit with your weapon (even though the rune is on somebody else's weapon)...but that would be rather pointless! (And doesn't seem to be the intent).