Though I never hid gifts because I'd just leave them at mom's (we'd always go to open gifts, eat dinner, and celebrate at her house), eons ago, when my sister and I were still youngish and thought there was a remote possibility of a Santa Claus and the Three Kings, like mamazita, my mother would wrap the gifts as she bought them, and she'd hide them in the closet behind luggage or behind the clothes racks on which the hangers with clothes would hang (probably the coat or dress area of the closet would be best, since these garments tend to hang pretty low and can serve as a curtain to cover the gifts).
There was an unwritten rule about not rummaging through mom's closet for clothes or anything without having her assist us, or she'd go off on us about making a mess of things. And honestly, what kid would think there's something for them in mom's closet? We assume it is full of shoes, clothes, luggage, and junk. Speaking of junk, you could pile some junk like shopping bags with random things, items from the garage, etc., to throw the kids off from thinking there's anything of value for them in the closet. Other gifts would stay in her car trunk or dad's (which I didn't have a key for, so there was that). Bigger gifts my dad would store in his office.
Mom would also use those "to and from" sticker tags (like these: https://thumb1.shutterstock.com/display_pic_with_logo/424.... Since you mentioned not being able to tell which gifts may belong to whom, I thought I would make mention of it. Have you never seen these stickers or used them? You can buy a whole pack, which includes two sheets or more, at the dollar store for just $1. I highly recommend them, you don't even need to write someone's full name. If you're worried your kids are nosey and will go through your closet and stumble upon the gifts, you could use their first name initial or use some sort of "code" symbol. A heart symbol could be the gifts for the kids, a star symbol could be the gifts for mom, a rhombus could be the gifts for hubby, etc., etc.