Gemini
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Sharing
With each of your saved conversation, you can click on More Options, and select "Share" to get a link that you can send the conversation to. These links are available for as long as the conversation exists, which makes it not a great long term solution if you're planning on having a lot of saved conversations. To manage that, I recommend saving the conversation to Google Docs.
Clean up
With each of your saved conversation, you can click on More Options, and select "Delete" to remove the conversation. I found that it is more efficient to delete from a mobile device. There, you can long press and click "Delete".