Interesting, but given that the stock has been so extensively reworked, or entirely replaced from the wrist back (note the joint), the configuration of the barrel and the type of sights fitted, I'd guess it was a target rifle rather than a sniper's issue.
The fact that there was no standardization on one type of rifle would also suggest target rifles for inter-service(?) competition etc. rather than sniper's use.
One can see a Parker Hale or A.J. Parker sight appears to have been fitted to the P14 before the present one - which is marked "ARSENAL TALLINN 1935".
Another question would be why the Estonians were converting such rifles to 7.62x54R when their standard service cartridge was .303BR ? Perhaps using up stocks of that cartridge in competition rather than their service cartridge held as war stocks?