When you receive an engagement announcement, do you have to send a gift?

There is no requirement to send a gift for any announcement of an happy occasion, like births, weddings, engagements etc. But it is the expected thing. However, people do just want to share they joy and may include people in the announcement list that they don't expect gifts from.

No, there is not a "have to" regarding sending a gift when you receive an engagement announcement. It is something totally optional. You may choose to send something and you may choose to wait to send something and you may choose to not send anything at all, except for a congratulatory card.

Whatever you feel comfortable in doing, will be the right thing for you to do.