Does anyone know if we can put up a

depends what you mean by a loggia - bbut its most likely a no.   

Modicasa is right.  We have a property in Le Marche - local commune - Tolentino.  We considered putting up a loggia by our swimming pool.  We are lucky we have a good friend whoe daughter is a geometra. She checked for us and we definately need permission - although don't know whether there is a fee to the commune involved in this as we have not got that far yet.   Remember that most commune's have aerial photos of their properties and can tell it one has changed from one year to the next.Also interestingly the geometra told us that we would have to use a builder that is registered for Taxes/IVA and prove this to the commune to complete the work - I assume to confirm we were not using someone on the 'black' .

Surely it depends on the construction? My understanding for our comune is if it's not classed as permanent (e.g. wood build), then you do not need permission? Only sure way to find out is to ask the commune or simply go ahead a build…

Wood s a permanent onstruction material it is how it is fixed that used to be the way around the problem, but that has changed.   A loggia to me is a seris of arches, forming a sort of covered area, in stone.   A sort of big posh porch.   That is a building an would need permission.   Your comune will require the builder to have the DURC as clombardelli says - which means you can only choose a builder who is up to date with all his INPS - which puts up the cost considerably.  You need todo it with a SCIA and all the necessary permissions, as it will take cubatura from your house and you have a limited amount.  Basically,  If it is going to provide a covered area of any sort it will need permission.  

A tent? Or what about a free standing shed?

For that kind of construction, you need a permission, even if it's made of wood.