I can give you some examples from LEW. In LEW, there is a city (Orussus) where any item in the SRD up to a specified GP value is available to players. This need not be roleplayed out.
I had a character in an adventure in LEW. The character undertook an obligation to go on the adventure. During the adventure, the character decided that he wanted to visit an NPC that he met during the adventure once that adventure (and his obligation) was concluded. That visit would involve considerable travel and would also expose the character to some danger. I concluded that I would need a DM for that, in the interest of fairness to all the other LEW players.
In-character, the character also reasoned that due to the risk of the post-adventure travel, he needed to travel with a larger group. I contacted the DM of the first adventure, stated my character's intentions, and asked if he would be willing to DM a sort of sequel to the adventure he was currently running. He graciously accepted.
The first adventure concluded in Orussus. The DM submitted an adventure proposal to the LEW judges for the sequel. During the downtime, my character went "shopping" in Orussus, and picked up a very nice special item (out of the SRD, but built in a "one from column A, two from column B" fashion). No judge was needed for that, since the item was right out of the SRD and within the stated GP limit for Orussus.
The sequel adventure was approved, and my character did the recruiting for the DM in the LEW tavern (recruiting) thread. A LEW Judge is judging the sequel adventure, even though both the DM and I are LEW Judges (we don't judge adventures that we either play in or DM, unless the DM drops out and the Judge has to take over...in which case another judge picks up where the first left off).
So, there's some flexibility and interpretation here. Just strolling around town...no big deal, just do it. Going on a 3-month trip through the wilderness to retrieve the Stone of Ar...get a DM and a Judge. Generally, if it appears on your character sheet as a result of leveling up, a character judge will look at it anyway, so you may not need to role-play that out. If in doubt, you can always ask in the Detail Discussion thread.