Finding transformer paramters
When finding parameters of a transformer, can you perform the OC test
  and the SC test on the same side? Or is it necessary to perform them
  on separate sides?


OC and SC tests can be made from any of the two windings, and are usually related to each winding voltages and the equipment available for testing.


  I am aware that by convention in the real world we perform the open
  circuit test on low voltage side and short circuit test on high
  voltage side. But can we do them both on the same side?


You can test the same side, or one on each side, as it is more appropriate to the facility used for testing. The following figure represents the equivalent circuit as viewed from the primary or secondary, assuming a voltage step-down transformer.



OC test

The result of this test allows you to determine the resistive component (mainly core losses) and the inductive component (due to magnetizing current), with respect to the winding tested.
From the equivalent circuit, \$R_c\$ and \$X_m\$ can be obtained from a OC test on the primary.

SC test

The equivalent circuit for the OC test:


with this test, you can determine the total resistive component \$R_s\$ (mainly copper losses) and total inductive component \$X_s\$.
The values obtained, as shown in the equivalent circuit corresponding to the sum of the values of the winding in which the test is run, and the reflected impedance values from the other winding.  

Considering impedance reflection, either test can be performed in either windings.
Normally, to SC test, you should have a bridge for low resistance measurement and determining the winding under test resistance and then, with this value, get another winding resistance (reflected).

How do you decide on which winding make the test?
in practice, this is determined by the equipment available for testing.
For example, performing the short-circuit on the low voltage side, has the advantage that the power supply of the high voltage side will not be of a very high current.