A good conversation does not have to end with an exchange of phone numbers. You can enjoy meeting someone and leave it there. If you would like to reconnect, keep it simple and respect their privacy.
If you share a public community space
You can continue the conversation in a group space where you both take part. Mention the event or shared interest without posting anyone's personal details. If the other person responds, let the conversation develop at a pace that feels comfortable to both of you.
If you see them again
Say hello and mention what you talked about. You could ask whether they would like to exchange contact details or meet at another event. Give them an easy way to say no.
If you do not cross paths again
That happens. Do not ask someone else to share another attendee's private information. If you have no shared way to reconnect, let the moment stand as a pleasant conversation.
For next time
If you would like to stay in touch, ask before the event ends:
“I enjoyed talking with you. Would you like to swap contact details or meet at another event?”
Either answer is fine. A connection should feel mutual.
For message ideas after exchanging details, read a simple follow-up you can make your own.