My partner and close friend who I’ve worked with for 20 years has moved to Panama. We are 50/50 partners. However, he’s paying much less to nothing in taxes and his cost of living is much less. My question: is there a precedent or protocol for fair payment in our situation?

In a partnership, profits can be split in any way the partners desire, provided both partners agree on the division. Several approaches exist to ameliorate the condition you've described, where there is a good cause for an uneven split in profits.
First, you could renegotiate the ownership split. Instead of a 50/50 stake between partners, the partnership could be set-up on a 60/40 or 70/30 basis.
Second, you could declare the difference in personal expenses as a specific value, and take that out of future revenue off the top. In other words, you would maintain the 50/50 partnership structure but pay each partner a base salary, with the Panamanian partner receiving a salary of $1 and the non-Panamanian partner receiving a salary equal to his increased costs of living. Profits above the salary amounts can then be split equally.
However, unequal distribution of profits in a partnership is usually a factor of unequal investment of time or talent, and not the inherently lower living expenses of one or the other partners, whether that's due to a more favorable tax situation in the chosen domicile, fewer dependents, or a personal decision to live a more frugal lifestyle. Unless there is a compelling and overriding business reason that requires one partner to live in a specific locale that is more expensive than that chosen by the other partner, it might difficult to make an argument justifying a higher share of profit.
Ultimately, any modification of the partnership agreement would need the consent of both partners, which may be difficult to achieve if it would have the effect of lowering the profits of one.