Buying land

I'm looking to buy a piece of land next to our house in Abruzzo but what happens if there are other names or livello on the plot, does the owner need to remove them before the sale?

Hello, you will find that it is quite common that multiple members of family own property and land here.Presumably left to remaining members. It is unfortunately quite common that this is never resolved but you may be lucky.
Your legal representative will try but in short yes, they all need to agree it.Never buy anything on a handshake.
Hope it goes well.

You will need a notary to assist you. They search out the owners and then you will need to get signatures from all of them. You could be lucky and discover there are only one or two names, or it could be a nightmare going back generations! The recommendation is to never buy property unless you have all the signatures or any one person can go back and claim the property and you are out of luck as that person has preference over you in court.

If you get a chance, you might read the book Pan' e Pomador by Ian McEwan. He and his wife wrote about exactly this subject. It is worth the read! There were like 15 people attached to the ruined villa they wanted to buy. They did end up getting it but it was really a frustrating challenge! Good luck!

A notary will not assist  you.  It is not his job to see if a property can be sold.   He is there to make the sale legal. 
You want to buy a piece of land which has more than one owner or has livellari?   If it has more than one owner then obviously the seller has to make sure they all are willing to sell - and/or alive.   If there are some dead ones, then he has to do the relevant successions to give the heirs the ownership.    Livellari is a different matter, and it depends on how they became so - they rented the land, they had usufrutto/enfiteusi?   It may just need updates at the catasto, or it may need a rettifica to sort out the problem.   However the seller needs clean title to sell.  And its his problem.  You can say you're interested, he can promise to sell - but if he cant fulfil his promise he takes the flak.  I can promise to sell you the Colisseum, but it doesnt mean I can.  The rule is not to hand over any money until the seller has clean title.

Greetings. You need a geometra (a kind of surveyor) to check the legalities. If there's more than one owner, they all have to agree, and any illegal structures, including gazebos, should be destroyed.