wastegate on the cold side ( on the intercooler)

What are the pros and cons from putting the wastegate only on the cold side instead on putting it on the hot side (pre-turbo)?.

It is less fuel efficient as you are demanding more turbine work and hence exhaust manifold pressure than using an exhaust side gate. For the same reason you probably won't achieve the same power as exhaust side wastegating on a knock limited fuel. There is more potential to overspeed a turbo unless it is well beyond your target power in terms of compressor pressure and flow capacity.

It obviously has potential packaging advantages. You could use it in a 2 stage approach if wanting to run for example an internally exhaust gated Borg Warner EFR 8374 on a ported 13B, these are known to boost creep from a low target. You could set a 3 port MAC valve or similar on the BOV reference line such that it sees atmospheric pressure when you only want low boost then operates as normal for higher boost targets where the exhaust wastegate flow is sufficient to maintain control. Likewise trying to progressively tune any turbo package from near atmospheric pressure at WOT it will potentially allow lower boost targets while establishing fuel and ignition mapping below what can be maintained with an exhaust wastegate spring which is significantly higher or having to swap aftermarket spring out or run a 4 port valve or compressed gas or an electric gate with the associated ecu/software costs for a wider control range.

It also provides a potential fail-safe other than a fuel or ignition cut if an exhaust gate system fails, meaning a vehicle could still be driven WOT at a lower power which could be useful in a racing situation or tow/high load application.