I know that the organization that distributes IP addresses decided to assign 192.168.xxx, 172.xxx and 10.xxx to private networks. However, I thought that private networks have their own address space so shouldn't a private network be able to assign any values in the IP address space and not be limited to those values? 

Assuming IPV4 CIDR notation

RFC 1918 allocates the following for private address space:

10.0.0.0/8
172.16.0.0/12 (not 172.0.0.0/8!!!)
192.168.0.0/16

While those are private, network engineers often use NAT to allow users on those nets to reach internet resources.
If you used 8.0.0.0/8 for private address space (for example), you would not be able to reach the google address server 8.8.8.8, because you would have an internal route for that block.
In addition, even if your “private” servers did not need to reach the internet at all, if google tried to reach your public webserver, and your public webserver had your internal routing table (with your “private” 8-net route), the replies would not get back to google.
So use the RFC1918 private address space and save yourself a bunch of trouble.
