Q:

How does a Potion of Poison work? 

A compatriot of mine and myself are in a disagreement about the way Potion of Poison works.  
He is a bit of a rules lawyer, and I trust his judgement, but I don't agree with his assessment of this item. 
He believes: Drink the potion, take 3d6 damage, roll the save.  On a success you are not poisoned and take no further damage.  Fail and you take 3d6 damage each turn until you make the save.  Once the save is successful the damage is reduced by 1d6 each turn until it is Zero. 
I believe: Drink the potion, take 3d6 damage, roll the save.  Succeed and the damage is reduced by 1d6 each turn until it is Zero.  Fail and you take 3d6 damage each turn until you succeed on the save, then the damage is reduced by 1d6 until it is Zero. 
It's a minor difference but a major effect.   We've read and re-read the description, coming to same conflict. 
How does a Potion of Poison actually function? 

A:

You have to be poisoned to take the damage (he is correct)
The description of the Potion of Poison says:

If you drink it, you take 3d6 poison damage, and you must succeed on a DC 13 Constitution saving throw or be poisoned.  At the start of each of your turns while you are poisoned in this way, you take 3d6 poison damage. [...] 

If you don't fail the initial save, you never get poisoned.  If you never get poisoned, you never take the additional damage and don't have to make another save and would only be hit with the 3d6 damage from drinking the potion initially. 
Another potential confusion point: you need multiple saves to reduce the damage to 0
You say that you and your friend interpret the item's effects to say:

Once the save is successful the damage is reduced by 1d6 each turn until it is Zero. 

and

each turn until you succeed on the save, then the damage is reduced by
1d6 until it is Zero. 

To me, that seems to imply that you both think that one save is enough to start the damage automatically reducing to 0 each turn.  However the item's description says:

At the end of each of your turns, you can repeat the saving throw.  On a successful save, the poison damage you take on your subsequent turns decreases by 1d6.  The poison ends when the damage decreases to 0. 

So with one save you can decrease the ongoing damage by 1d6.  So, if you make a save while poisoned and taking the full 3d6 damage, that ongoing damage will be reduced by 1d6 to 2d6  damage which you will take every turn thereafter until you make another successful save.  Essentially, you are going to need to make 3 saves to end the poison. 

A:

Passing the initial save means you only take the first 3d6 poison damage (your friend is correct)
Note the description of Potion of Poison states (emphasis mine):

If you drink it, you take 3d6 poison damage, and you must succeed on a DC 13 Constitution saving throw or be poisoned.  At the start of each of your turns while you are poisoned in this way, you take 3d6 poison damage. 

As you can see, succeeding on the first save means that you are not poisoned.  poisoned, in this case, refers to the specific condition (not the same as taking poison damage).  The damage taken on subsequent turns only happens while you are suffering from the poisoned condition so if you are not poisoned by succeeding on the first save, no further damage is taken. 

A:

He is correct. 

"If you drink it, you take 3d6 poison damage, and you must succeed on
  a DC 13 Constitution  saving throw or be poisoned." 

The first save is to see if you become poisoned. 
If you make this Save, you are not poisoned, and take no further damage. 
If you fail, you must then make a Save each round to see if you can overcome the poison.  A success reduces the Damage-per-Turn until reduced to 0.