In the broadest sense, any person can do any puja any time. Just as any driver can drive any vehicle, with or without training, or a license, as long as they know a few prerequisites like turning a key, etc.
At the same time,at times initiation is necessary.To be able to consecrate a temple training is needed and initiation is prerequisite step.Without training no installation can happen and without initiation there is no training.For specific kind of pujas(which are usually advanced/complex/tradition specific in nature) additional knowledge is needed but general puja is simple and undemanding.Only a professional can handle a truck or fly an airplane however no license is needed to ride a bicycle as it is simpler than the rest.
Also it depends partly on what kind of Moksha the sadhaka is aiming for.(I'll continue the rest of the post in Shaivism Discussion thread as putting it here would be digressing from the OP.)
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