I don't know if I'm crazy or what, but if you had asked me "Do you need a free hand to Shove?" I would answered with a resounding of course before I reread the PHB and noticed it said nothing of the sort. It specifically mentions you need a free hand for grappling, you need a free hand for somatic components (feats notwithstanding) and well, seems to me like adventurers cannot normally juggle more than one thing per hand. Is this an oversight for a tacit rule (therefore yes, you can only do one thing with each hand, and wielding a weapon prevents any other use), or there's a rational reason for which you can shove a creature when you are duel-wielding daggers I'm missing, or does it make sense in-game for balancing reasons I can't see...? Help me out here
No, you do not need a free hand to shove
Your own research into the description of the special Shove attack indicates that it does not require a free hand, while other special attack actions do explicitly say you need a free hand. That's pretty good evidence that shoving is possible even with your hands full.
You could shove someone without using an open palm-- shoving with a fist, holding a weapon or otherwise, seems like it would generally work well enough. It's also perfectly rational that you could shove without using a hand at all, free or otherwise. You could shove someone by ramming your shoulder into them, for example. You could shove someone with a "this is Sparta!"-type forward kick.
I don't see any reason why you might need an open hand or grip to shove a creature or object, though edge cases certainly may exist.
8\$\begingroup\$ Other narrative explanations include pushing with a shield (good for not getting stabbed as you close with them), or a "sweep the leg" takedown if we're talking about shoving someone prone. But shoulder-check was the first thing that came to mind for me. Or just rush someone and cannonball into them. \$\endgroup\$ Mar 4, 2021 at 8:58
\$\begingroup\$ This is Sparta!! (gif) (source) [Feel free to edit this into your answer and flag this comment as no longer needed.] \$\endgroup\$– MakyenMar 5, 2021 at 8:36
\$\begingroup\$ BUMshove stranger into waitress is a good way to start a bar brawl \$\endgroup\$ Mar 5, 2021 at 10:25
You don't need a free hand to shove
Why? One reason is the lack of an explicit rule requiring a free hand. Grappling, on the other hand, has a rule explicitly requiring that free hand.
Using at least one free hand, you try to seize the target by making a grapple check, a Strength (Athletics) check contested ... (Basic Rules, p. 77)
Another reason is that you can look at a related rule, for unarmed strikes, and notice that various parts of the body can be used for that kind of non-weapon attack.
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). (Basic Rules, p. 76)
A hip check/bump (has happened to me on a dance floor IRL), lift your foot and shove, shoulder lowered and drive - all of these could knock an opponent sideways. Or even knock them prone. (I was knocked prone by body blocks when playing football more times than I can count).
Given that a Shove is a special use of the Attack action (CH 9, Combat), it is reasonable to apply that same logic used for unarmed strikes to a Shove - which it is very closely related to since it is a non-weapon use of an Attack action.
Shoving a Creature
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you’re able to make multiple attacks with the Attack action, this attack replaces one of them. (Basic Rules, p. 77)
a. Shove is a special use of the attack action
b. Shove normally doesn't involve a weapon (Shield Master feat being an explicit exception)
2\$\begingroup\$ Given how 5E is usually consistent with how it applies "specific > general", the parenthesis mentioning the "shield master" feat is what seals it for me. The fact that the rules specify that the shove from shield master must be done with the shield tells me that if the "general" shove was intended to require a free hand, the rules would likely say so explicitly. And now I can describe my "shove to prone" attacks as a Zidane headbutt. Awesome! \$\endgroup\$– DungarthMar 3, 2021 at 17:15
\$\begingroup\$ @Dungarth Oh, yeah, complete with youtube video! 🤣 \$\endgroup\$ Mar 3, 2021 at 17:56