Depending on how realistic you wish to play, this background info may help:
From a martial (HEMA) perspective, disarms (unless slippery-fingers-accidental) usually involve you using your off-hand to grasp the weapon (typically grabbing the opponent's hilt). You can do this easily with a shield (or even buckler – a hand-held shield) in your hand, though it may reduce the effectiveness of the shield as defence. You would usually be left holding their weapon awkwardly (so you couldn't easily make an attack with it – I'd say it'd cost you an action to move it round to an effective grip).
I believe you couldn't easily stash someone else's weapon safely – swords and knives usually only fit in their own scabbards, and shoving a sharp weapon into your belt is likely to leave you belt-less. That said, you could probably drop a dagger into an accessible pouch.
However, you can easily deprive them of it by throwing it or dropping it and kicking it away. If they drop their weapon, you're much better off kicking it away or knocking it aside with your swordtip, than attempting to pick it up off the floor. Picking a weapon off the floor would (in reality) cost you an 'action' of time, and leave you very vulnerable whilst you did so – I'd say the opponent would get an opportunity attack, if they have another weapon to hand, as you're essentially disengaging from the fight to pick something up from the floor. Cool rolls across the floor grabbing a weapon as you go are choreography, not martial.
I've seen fighters disarm and kick a weapon away.
I've only ever seen someone successfully pick up a weapon off the floor when their opponent was either well out of range, or laughing so much about the disarm that they stopped paying attention to the fight. (And yes, that can be done deliberately... throw your sword at someone and whilst they're confused, stab them with your dagger can work... but only once).
Note that this applies specifically to bladed weapons on the floor. Things on the floor are much less accessible than things at or above waist level, as you have to stoop or crouch to reach them. Weapons are particularly hard to pick up, as you want to grasp them correctly (not by the sharp end!). Grabbing a weapon carelessly with an ungloved hand risks injury. Grabbing a weapon with a gloved hand is difficult – getting big gloves / gauntlets into hilts can be tricky depending on the weapon.
Picking up a pole weapon would be much easier, and I think I've seen it done during combat; with pole weapons you're also at a slightly longer range, and it's a little harder to make a sudden reaction than with say a rapier, so you're a little safer from an opportunity attack.