I understand the purpose of both but just verifying if this correlates together at all.
Private addresses are address ranges set aside specifically for not being used on the Internet by IANA. Anyone can do anything with these addresses except using them on the public Internet.
Subnetting is the process of splitting one address range into two or more smaller ones that are contained within the intial range. Check the excellent answer to this question for how this works.
Both of these are completely unrelated. You can subnet any public or private address range that is large enough. The result doesn't change the public/private property of the initial range.
Private IPv4 addressing was chosen (somewhat arbitrarily) by the IETF with the cooperation of IANA. It is detailed in RFC 1918, Address Allocation for Private Internets.
IP, itself, has no concept of private or public IP addressing. Private addressing is simply the ISPs adhering to RFC 1918, and agreeing to not route any traffic for the RFC 1918 addresses.
Private addresses come from three specific ranges of IP addresses specifically reserved for that purpose - to be used on an internal network. They are completely unusable as a public address, as there are potentially thousands of devices having that specific IP as part of their separate network. Communication between those can only happen through public addresses, which are like an "entry point" from outside to anything from a single computer to a complex combination of internal networks, and the request will then navigate through those and end up at a specific device with an internal, private address. There are devices that perform this "conversion", it is called Network Address Translation, or NAT.
For example, if you are on your home WiFi, your PC might have an IP of say 192.168.1.101 - this is in private range. You request the some website, which has a (public) IP of say 22.214.171.124. Your router takes your request, and sends it to wherever it needs to go to reach 126.96.36.199, and makes a note that the return address is actually 188.8.131.52 (that's the router's IP on the Internet, sometimes this translation happens multiple times). When the request arrives at its destination, it also ends up at a router most of the time. This one determines which IP in its internal network the request should actually reach, and forwards it to the private, internal address of the actual webserver (which can very well also be 192.168.1.101). Server then responds with a message destined for 184.108.40.206, the public address of your router (or some other exit point if multiple translations happened). When the response gets there, the router then determines which device originally sent the request and forwards it there.
As far as subnetting goes, this commonly happens with the private ranges, because often networks don't need that many addresses, but more than 3 networks are needed (for multiple reasons like separation of different departments or device types, as well as security).
For example, the 10.0.0.0 range is a /8 range (only the first octet is constant), so that's over 16 million addresses. Often, smaller sections are carved with subnetting, commonly, /24. Then it is possible to have a 10.0.0.0 network, then a 10.0.1.0 and so on.
TL;DR: Private addresses are just specific ranges defined to be only for internal use, allowing the same subset of addresses to be reused again and again to allow more individual devices without exhausting the IP address range. Those private ranges are often subnetted to be able to have many separate networks.
Private IP addressing pools are similar to public IP pools. It in not like that private pools are derived from public pools. Public and private pools are result of IP subnetting of 32 bit IP. Then, it is understood between the ISPs of the world that they will not advertise private pools from their network towards internet. On the other hand, public pools will propogate through internet and there will be no duplication between public IPs.