I was pondering what they had in mind for these Public Companies and I came up with a few ideas. First off, I'm fairly sure these new companies will fulfill the need for competition between the 3 main cities and perhaps even their surrounding hamlets. Incorporating that previous statement, it could even become possible to claim the hamlets as part of your territory for the company you represent. This would be similar to obtaining outposts in FFXI, once you obtain enough influence in a region you control the Hamlet for that area. They could add the ability to warp to areas under the control of your company as well as the use of the small guild shops within those hamlets.
The public companies are also apparently supposed to be instrumental in dealing with the coming threat to the peace of Eorzea, as such I imagine they will be involved in company specific quests and battles you can undertake and hopefully gain you some manner of currency that you could use to trade in for unique items or services. They could even work in some manner of PVP battles that competing companies could engage in to vie for territory. (Though the only PVP I want to see in the game is Triple Triad) . Alternatively, in lieu of battling one another, two teams from competing companies could battle the same targets and fight to complete a job before the other team. Such an idea would serve to reward the team with better communication and teamwork assuming it was instanced and both teams started on equal footing.
Crafter and gatherers should be in the loop as well with quests or supply requests to obtain items for the NPC's within the company or outfit them with new gear. Though I'm struggling with coming up with an idea to implement that would differ from being basically a guildleve with more lore/NPC interaction behind it. Though gathering or crafting unique items or working together on larger scale items would be a start. This would still allow non-battle classes to gain rank/points within the company. Then entire system wouldn't largely differ from how Campaign (at least the Ops portion) worked in FFXI.
As for private player run companies, I imagine that is still in the works but I hope it still involves player housing/boats/gardens/etc. Those alone would serve to keep me busy for weeks/months/years on end working to improve my companies resources and facilities.
I'm interested in your thoughts and what others ideas/concepts you would like to see implemented as well.