Being redirected to the signup page after payment is the default. Do you mean you aren’t getting any message that the payment was successful?
As to your question about HTTPS, that IS possible. PayPal has been phasing in requiring HTTPS for all transactions: including buttons (which has never been the case). Do you have s2Member logging enabled? If so, check the logs with “paypal” in the name for error messages (or open a support ticket and let us check them for you).
I thought I might have an answer for your original problem, but if you are not using HTTPS then I don’t have one off the top of my head.