If neither player controls "The Arena's Greatest" on the battlefield, does the effect activate or no?

Yes, the effect will still activate.

According to Rule 187.6.b, when a battlefield is uncontrolled, its abilities are also uncontrolled. The turn player takes responsibility for adding the ability to the chain and is treated as its controller for the purpose of the effect.

As stated in the rule's example:

"Example: The Arena’s Greatest is a battlefield that reads 'At the start of each player's first Beginning Phase, that player gains 1 point.' This ability will usually trigger while the battlefield has no controller. If it does, the Turn Player goes through the steps of adding the ability to the chain and receives priority after doing so, exactly as if they controlled the ability."