While in a friendly or neutral settlement, the player should be able to allow companions to disperse and interact with the site and its inhabitants. This would give companions the chance to fulfill needs, obtain equipment, and play a bigger role in the world, rather than serving solely as ablative meat for the player.
Dispersing the party temporarily disbands it, its members agreeing to return to the current location and continue travelling together at a set time in the future. From the player's perspective, this then cuts to a screen like the one used for waiting and sleeping, with the adventurer either waiting at the rendezvous point or wandering off to interact with the site like their companions.
In the latter case, the player should be able to choose beforehand what the adventurer does in this time period. This would allow them to perform actions that aren't currently available in adventurer mode by handling them off-screen, such as receiving medical attention or satisfying otherwise unfulfillable needs, such as making romance.
After dispersing, companions should be free to do as they please, acting independently in accordance to their needs and preferences. For instance, companions with a desperate urge to get drunk should be more likely to prioritize a visit to the nearest tavern, but if the site lacks such a location, they should attend to other matters.
One such matter would be to visit the local shops in order to replenish ammunition and replace damaged or destroyed equipment. Apart from a few obvious restrictions, chiefly "do not trade away artifacts" and "do not replace equipped weapons and armor with inferior metals", this should be a hands-off process, with the only influence the player exerts being to provide their companions money or items to trade with beforehand. The end result of a shopping trip should reflect both the companion's tastes and the site's available wares, but as a rule the companion should come out of it better off than when they went in.
Regardless of what they actually do during this time period,the party should also be socializing with the local population, both to satisfy that need and to gather and spread rumors as they go about their business.
All of this should be affected by how much time the companions have to act on their own. Too little time should limit what can be done, whereas too much time leaves the player waiting for a timer to count down. Going back to the tavern example, if the party has dispersed for a very short period of time, a companion may simply go over, have a drink, and come right back. However, if the site's nearest tavern is too far away to get to and back from in the given span of time, they may opt to do something they can actually get done and be back from in time for the rendezvous. On the other hand, giving the companion too much time may lead to them spending the night at their chosen watering hole, provided it has a free bed.
Whatever happens, once time is up, the party regroups and the player recieves a summary of the each companion's experiences while they were away, similar to mission reports in fortress mode. This should include information such as what they did, any items they acquired or traded away, new rumors, and any combat they witnessed or were involved in. Once that's done, gameplay resumes as normal and the player is free to continue.