Most of the time, when you think of throwing a weapon like an axe or a spear, it's done with one hand. However, some weapons like the trident are versatile... is there anything that says you can or can't throw the trident two-handed for 1d8 instead of 1d6 damage? I guess for a blunt thrown weapon you could do it like the olympic hammer toss, but that doesn't apply with things like the trident.
From a balance point of view, you will have paid the cost (of not having a shield or second weapon equipped in your off hand). So that's OK.
In terms of rationale, watch someone throwing a javelin. The off hand is used for balance and additional momentum. There is definitely more force when the off hand is free*. So that's OK too.
I'd allow it.
Note that is not a RAW interpretation, due to the wording of the Versatile property specifically mentioning improved damage to a melee attack only. I may still allow it, so will leave the answer here. As far as I can see the melee-attack only constraint is in v0.3 of the online rules, but I cannot search further back. If anyone has older versions and can verify whether/when the wording has changed, I am happy to update the caveat just for accuracy's sake.
* yes this is a little facile. Don't look too hard at the narrative argument. Don't sweat the rationale to the point that a degree in Biomechanics is required. Don't ask me about hit points :-)
I would say no and here is why:
Thrown. If the weapon has the thrown property, you can throw the weapon at a target to make a ranged attack. ... Versatile. This weapon can be used with one or two hands. […] when the weapon is used with two hands to make a meele atack
Thus being Throw a ranged atack Versatile (meant for meele atacks) would not apply.
However as aforementioned if the player can make it sound right and it would make us enjoy the game more I would allow it.
By RAW, I'd say it is ok if the weapon is specifically thrown versatile. Without penalty.
Thrown. If the weapon has the thrown property, you can throw the weapon at a target to make a ranged attack. ...
Versatile. This weapon can be used with one or two hands. ...
RAI I really don't know. Balance wise, I think requiring both hands is cost sufficient for the improved damage dice which only average out to 1 damage.
Go outside, find a broom or something else with a long shaft. Try and throw it with 2 hands. It is near imposible for you to coordinate a throw with both hands and also hit something. And also most of the time one of the hands will break the speed insted of increasing it.
Is it like an axe or hammer, where you can with both hands raise it over your head, and then throw it with both hands, it is another thing. (But you did mention trident so thats what i thought for reference)
This is where you need to ask your self, does the rules count or does realism count? Almost always, if it sounds plausible, if the player can argument (he is a 1000BC ninja with roots in the special two handed trident diciplin that he have been training for his entire life - ofcause he can), or if it is possible in real life, id allow it. Else, id allow it with a penalty, penalty varys depending on dificulty.