What happens when two planeswalkers with the same planeswalker type are controlled by different players?

As of the Magic 2019 rules update (July 2018), the 'planeswalker uniqueness rule' no longer exists. There is no rule that causes a planeswalker to be put into a graveyard simply because another player controls a planeswalker with the same type. Each player's permanents are evaluated independently for the legend rule and similar restrictions.

The legend rule (CR 704.5j) only applies to legendary permanents controlled by the same player. Planeswalkers are not legendary by default (though most modern ones are printed with the Legendary supertype), so even the legend rule only matters within a single player's board state, not across opponents.

If two planeswalkers share the same planeswalker subtype but are controlled by different players, both simply remain on the battlefield and function normally. There is no state-based action triggered by this situation (CR 704.5j, CR 306).

Concrete example: Alice controls Jace, the Mind Sculptor (a Jace planeswalker) and Bob also controls a Jace, Wielder of Mysteries. Both planeswalkers stay on the battlefield simultaneously with no issues — neither player is forced to put one into a graveyard.

Note: If the same player somehow comes to control two legendary planeswalkers with the same name or two with the same planeswalker type (and both are legendary), the legend rule (CR 704.5j) would require that player to put all but one into their graveyard. But across different players, no such rule applies.