The idea behind the private share is that it is set to private and only accessible by accredited IP's which you list in your config. As far as I know NFS doesn't support users.
root@MIMNAS:~# exportfs -v
/mnt/user/MIMPVE-4TB
192.168.1.150(rw,async,wdelay,hide,no_subtree_check,fsid=102,sec=sys,secure,root_squash,no_all_squash)
/mnt/user/MIMPVE-4TB
192.168.1.99(rw,async,wdelay,hide,no_subtree_check,fsid=102,sec=sys,secure,root_squash,no_all_squash)
/mnt/user/MIMPVE-4TB
192.168.1.24(rw,async,wdelay,hide,no_subtree_check,fsid=102,sec=sys,secure,root_squash,no_all_squash)
/mnt/user/MIMPVE-4TB
192.168.1.100(rw,async,wdelay,hide,no_subtree_check,fsid=102,sec=sys,secure,root_squash,no_all_squash)
Thats what the output looks like when you set it to private, my rule was: 192.168.1.121(sec=sys,rw) 192.168.1.150(sec=sys,rw) 192.168.1.100(sec=sys,rw) 192.168.1.99(sec=sys,rw)