I've privately shared a track but the recipient didn't get a notification?

They may not have received the notification because:

a) they've turned their email notifications off. They can turn these back on in their email notification settings.
b) the notification may have filtered into their junk or spam folder. They should check to see if this happened and if it did, get them to add notifications [at] soundcloudmail [dot] com to their safe senders list.

Alternatively, you may:

a) set up a secret link to the track and share it directly (now, available for all accounts).
b) have them go to your profile page and listen to it (make sure they're logged in first.)