Being a Story Listener can sometimes make you emotional because you can identify with a story or just feel very sad for the Story Teller. This is okay; you are only human. It can make you feel uncomfortable, but the Story Teller may not mind you being emotional. It will likely only benefit your relationship because it shows your empathy and commitment. Everyone is emotional at times and you are allowed to express your feelings and emotions.
There is no deep conversation without feelings. We have empathy for a reason; it promotes contact and understanding.
You can however prepare yourself for these situations by identifying your own emotional triggers. What experiences in your life make you emotional?
If you know in advance what topic the Story Teller wants to explore during your next session, think about whether you're comfortable talking about that topic or not. If not, let the Story Teller know so they are aware this is a sensitive topic for you.
Express how you feel
When you become emotional during the conversation, tell the Story Teller that this topic moves you.
Do not get overwhelmed
If you are overcome by your own emotions and it is too much to handle, you can take a time out or end the session.