I guess my brain doesn't work the way other peoples do, in your scenario. I wouldn't be frightened to then see something terrible, I would be like "well that explains it". I guess that situation doesn't really instill fear in me. I was in New Orleans after Katrina (3 days later, to take photographs of commercial property that was damaged), there were still bodies floating in the street and plenty of atmosphere others described as creepy, to me, it was exactly what I expected, there weren't any abnormal feelings about it. That said I'm not unscareable, even some movies come to mind that genuinely scared me (not jump scares either), but it was more a situation of utter and complete helplessness that makes me feel fright than anything else.

Most suggestions so far are plot-related and based on the premise of counteracting the inherent comfort of having the support of a team by making the game more similar to a single player one: ignorance and distrust between partners, forced splitting of the party, betrayal.
There is, however, a specific way in which a cooperative game can be more frightening, or at least more worrying, than a single player one: fear of being hurt by teammates.

• An idiot opens the wrong door at the wrong time.
  
• Someone greedy steals weapons etc. you need.
  
• You could be stealthy if you were alone, but there isn't enough cover for two.
  
• Plenty of grenades and, generally, other splash damage and hard to aim indirect attacks. Are you safe from eager or fumbling friends?
  
• Traps, triggered by one and affecting the whole group.
  
• Is your stranded or lagging companion worth rescuing or waiting for? Climbing out of a pit can take enough to be reached by monsters.
  

There is, however, a specific way in which a cooperative game can be more frightening, or at least more worrying, than a single player one: fear of being hurt by teammates.

• An idiot opens the wrong door at the wrong time.
  
• Someone greedy steals weapons etc. you need.
  
• You could be stealthy if you were alone, but there isn't enough cover for two.
  
• Plenty of grenades and, generally, other splash damage and hard to aim indirect attacks. Are you safe from eager or fumbling friends?
  
• Traps, triggered by one and affecting the whole group.
  
• Is your stranded or lagging companion worth rescuing or waiting for? Climbing out of a pit can take enough to be reached by monsters.