Private IP and public IP
A private IP is not necessarily assigned by an ISP. It is actually a much more general concept: Any node (or cluster of nodes) realizing NAT separates two networks, a public and a private one. Most routers in home networks separate your internal network (private) from your ISP's or the wider internet (public). In the case of an ISP, it might be so that the ISP uses NAT to separate its own network of subscribers (private) from the wider internet (public), though not all ISPs have such a policy; actually some ISPs may even provide you with a reserved public IP (i.e., an IP which is accessible from the wider internet and which is statically assigned to you).

A private IP is only valid within its own network. In fact, the values for private IPs are standardized and are not routed in the wider internet. Thus, even if a node (e.g., a webserver) in the internet came to know the IP of your device in its private network—which, up to exploits, is unfeasible because of NAT—, it cannot really do anything with that IP unless it is also part of the device's private network: any packets sent to that IP will be immediately dropped (i.e., not routed) by the next node. Rather, what is logged in webservers and the like is the public IP of the NAT node through which you are accessing it.