When an user turn on Internet, a PDP session is created and an IP is assigned with the help of GGSN, which maintains session logs and does the charging with the help of PCRF. Mostly, Mobile users will get private IPs. Public IPs are rarely assigned to mobile customers due to shortage of available pools. Whether users shall receive private or public IP, all that configuration is done on GGSN. From GGSN, traffic is pushed towards ISPs. On the edge router, which is termed as Internet Gateway, it performed NAT44 or NAT46 (Depends) and then routed towards ISP network. 

In short, Private IPs are assigned to customers with the help of GGSN and NAT is done edge routers. Read about CGNAT, these days every ISP is implementing it to NAT millions of IPs all together. Happy Learning!