They will send you a letter in the mail with your sponsorship decision. It depends on which Visa office you're going through what you will hear next. Typically, the next step is an AOR, or Aknowledgement of Receipt from your Visa Office. For the Decemeber group, this has been taking the form of an update on eCAS that says your application and medicals have been received. Some people in our group skipped the AOR and have gone on to "In Process" which is where eCAS is updated to say that your application is currently being worked on. We have three now who have a PassPort Request, and skipped the AOR and In Process stages completely.
In your sponsorship decision letter, it will tell you if you have met the requitements for eligibility to be a sponsor. It will then tell you which visa office your application has been forwarded to.
My letter said "While the Application for Permanent Residence has been found to meet minimum requirements for completeness under the Immigration Act, the visa office may, where necessary, request additional documents and information from your relative(s)." I would guess that if they noticed your application was missing something, this is where they would request that you send it.
And, lastly, it confirmed the receipt of the processing fee.
I'm not sure how Buffalo is doing. No one in the December group has heard anything from Buffalo. Someone in the November thread called them, and they said they were working on the applications they had received from Mississauga on December 1st.