The spell obscuring mist makes a "misty vapor" that is also called "fog".
Presumably this is "magic fog" as opposed to mundane fog, although I cannot find anything in the conjuration rules that specifically declares it.
A fourth-level Tempest Druid can see partly through it.
At 4th level, a tempest druid can see through 10 feet of magical fog, mist, gas, wind, rain, or similar inclement weather conditions, ignoring any concealment it might grant. This distance increases by 5 feet for every 4 levels beyond 4th.
A Cloud Gazer Sylph has a carefully worded ability to see through magical fog.
You can see through fog, mist, and clouds, without penalty, ignoring any cover or concealment bonuses from such effects. If the effect is created by magic, this feat instead triples the distance you can see without penalty.
What about a Shaman using either of the relevant two spirits?
Gaze of Flames (Su): The shaman sees through fire, fog, and smoke without penalty as long as there is enough light to otherwise allow her to see normally.
Water Sight (Su): The shaman sees through fog and mist without penalty as long as there is enough light to otherwise allow her to see normally.
Can the Shaman see though the obscuring mist because it is a kind of fog?
Or is the Shaman unable to see though the obscuring mist because the fog has a magical source?
Lacking the specific wording of the fist two PC abilities, I could understand either interpretation.
(The Burn Rider Barbarian and Flame Dancer Bard also want to know your answer.)
I think yes, all those can see through it.
Honestly don't know. I would agree normally; Yes they can see through it. Though historically there's almost always a differentiation between mundane and magical sources...
Some, like the ifrit's Firesight, simply don't mention it either. Enjoy your Ifrit assassin w/ smoke bombs, or an Eversmoking Bottle.