Can provide a little more information on what has been interpreted differently?
Conversations related to work. Along the lines of one giving details to two co-workers.
How long after the event/incident did you realize there was a difference in interpretation?
Varies. One hour to one week.
There are many, many internal processes that could cause and contribute to a person arriving at a different interpretation of the same event.
They may have a different sensory experience of the event. They may hear something different or not hear something important.
They may have a different memory for the same event, so when they communicate their memory it differs from another persons.
They may communicate their interpretation in a different way. So, they may have the same interpretation but the way in which they describe it may differ.
They may make a completely different interpretation of the same information due to the influence of emotions, memories and personality style (I think this is the one you are actually interested in). We rarely (if ever) interpret information in a robotic like manner where we only process factual information. Everything we do is heavily effected by our own emotions, thoughts, beliefs and experiences. A good (and simple) example, of this is the observed effect of low mood on memory recall - a depressed person will demonstrate a significant negative bias in the information they recall. This kind of 'filtering' occurs constantly. We attend to, select, member and value specific information in light of our life experiences. For example, a person who has been seriously traumatized will interpret latter events relative to their trauma. Traumatized individuals often attend to/notice/recall threat related information much more intensely than non-traumatized individuals.
So, assuming that the difference in interpretation does not reflect a different sensory experience or memory function then the difference in interpretation is likely due to the vastly different thoughts, beliefs, emotions, experiences that impact on any two peoples' interpretation of an event.
I hope this has been of some help. Please let me know if you have further questions or would like me to clarify any pat of my answer.