There is ambiguity in the construction "interview with".
Many English verbs participate in constructions along with prepositions or adverbs, and those are an important component of the meaning. With in this case simply means that both parties were participants in the interview. It does not further specify who is the interviewer and who is the interviewee.
If it were important to specify - for example, if it were not clear from context - you could use other prepositions.
If you were the interviewer:
Your interview of Barack Obama.
If you were the interviewee:
Your interview by Barack Obama.