No. One subscription covers everyone on your account. The people you invite get the same Premium features without buying their own.
The person who owns the account and sends the invites needs the subscription. A subscription bought by someone you invited doesn't cover your account.
The people you invite keep Premium while your subscription is active and they are still on your account. If your subscription ends, they lose the Premium features, and you keep everything they logged. Nothing is deleted. If you remove them from your account, they lose access to it.
If you have trouble inviting someone, open Help in the app and tap Contact us.