On a personal account, a link is a snapshot. It shows the conversation as it was when you made the link, and messages you add afterwards aren’t included. To share the newer version, open the chat, press Share and choose Update link.
A link can also include images and files you uploaded in the chat. Links are anonymous by default, but anything you typed is visible, names and personal details included. OpenAI notes that some older ways of sharing may show your name, so look at the preview before you send it.
You can also share a single answer rather than the whole conversation, where ChatGPT offers it.
If someone carries on the conversation from your link, ChatGPT makes them a separate, private copy in their own account. Nothing they write is added to yours.