For guests with no email address — and for anyone you would rather text personally — the send screen gives you their invite to send yourself.
Tap a person and your phone opens a message to them with the invitation link already in it. Send it, come back, and tap the next one. Each person gets their own link, which is what lets us tell their answer from anyone else's.
On a computer, use Share to copy a link or open the share sheet, and paste it wherever you talk to that person.
Until you actually send it, that row reads yours to share on your event page. We will not tell you an invite went out when it is still sitting in your hand.
Do not forward one guest's link to someone else — a link belongs to the person it was made for, and answers on it come back under their name. Use Invite more to add the new person instead.