The function of the paladin on his hands says:
Your blessed touch can heal wounds. […] As an action, you can hit a creature and take advantage of the pool to restore a lot of health for that creature, up to the maximum amount left in your mana pool.
It states no restrictions as to how you should touch them or if you need a free hand to do it. The name of the feature does not necessarily impose a restriction on its operation.
Likewise, spell casting rules say about touch-sensitive spells:
Most spells have a range expressed in feet. Some spells can only target a creature (including you) that you touch. Other spells, such as shield spell, affect you only. These spells have a range of self.
That's all it says about touchscreen spells. The only relevant restrictions would be those that appear in the description of a spell or feature itself, as there are no general rules for that. One can simply understand that any form of touch is sufficient for a touchscreen spell.
5th Rule Designer Jeremy Crawford unofficially confirms this interpretation on Twitter:
Does the Paladin's Rest on Hands feature require a "free hand"? Or can they use it as long as they are close enough to be affected?
The imposition of hands forces you to hit the target. The feature is not about how you run the touch.
Good to know I can ignore the title of the [feature] to determine how it works.
You are invited to play the name of the feature to the letter. In this case, I recommend you put both hands on the target.
Most groups play with a function like Lay on Hands with the name in mind: put on a hand. Yet the rules do not say what touch implies
Because "touch" does not have a special definition in the game, we rely on the general English definition of the word:
Establish physical contact with; put in contact the hand, the finger or another part of the body.
This definition does not require that we specifically use his hand to do it. As a result, any physical contact with one's body is sufficient. (As always, the deputy minister is free to decide otherwise.)