You can send posts you see in feed as a direct message. Messages won't appear in feed, search or your profile.
To send posts you see in feed as a direct message:

Instagram app for Android and iPhone
- Tap
below the post you'd like to send. - Select the person or group (up to 32 people) from the list below or tap Search to search for someone.
- Add an optional message at the bottom, then tap Send.
Note: When you send a private account's post as a message, only people who follow that account will be able to see the post. The person who uploaded the post won't be notified.