PayPal buyers will always be opted-in, so those buyers' info should always be sent to Aweber if you have it set up. Other checkout methods may support an opt-in/-out, so in those cases the buyer would need to opt-in for their info to be sent to Aweber.
When the buyer's info is sent to Aweber, they should send the buyer an email asking them to confirm their subscription. If that message doesn't arrive, the recipient should check their spam folder in case it wound up in there.