Export phones from USA. Transit in Italy

The USA does not mind. You buy the phones, you pay what is needed and put them in your luggage. TSA, the airport security people will ask you to put all phones through the scanners, no problems there.
4 phones is many but there is no legal limit. Carry-on luggage is the right place, not hold luggage, due to batteries in the phones.

In Rome, if you do not need to collect your bags and hand them in for the new flight, you will likely not see customs. Immigration are the people who will ask you for your passport, customs are the people who check whether you bring in illigal items, or need to pay a tax on it.

When you arrive in Albania you will need to go through customs, this is where you run the most risks.
There will be a maximum amount of value you may import into the country without having to pay import tax and 4 new phones are very likely over the limits.
Four second hand (or one new and three second hand) phones are more likely acceptable.
Keep your paperwork from when you bought them, showing the price you actually paid. Based on this and on what the customs people think they are worth they will work out whether you can bring them with you or have to pay.

Best is to go to the 'I have something to declare' line and ask the officers. It makes sure you have to pay if you bring too much but you will not get into problems.
The more risky thing is to just walk through the 'nothing to declare' line and hope they do not ask you.
If they do check you, you can be sure that all 4 phones will be found, also the two in your pockets. And in that case you will likely need to pay a fine and might have to hand it the ones that are over the limits.

Before you buy the phones you should look into the amount you are allowed to bring into the country.

You are legally required to submit a declaration of goods exported from the US that exceed $2,500 (USD) in value, including hand-carried goods (with some exceptions that probably do not apply to your case). It's even possible they could consider it a dual-use technology which might cause some hiccups.

The method of submission is either a form called an SED or an electronic AES submission. They are phasing in the latter for all modes of transport during this year (2016) as I understand it.

While outgoing customs checks from the US, as in most countries, are rare, I would always suggest trying to follow the rules. For example, CBP has been known to go after Canadian visitors picking up personal-use ammo for export at retail in the US, which neither the shops nor Canada Customs has any problem with (assuming the purchaser has the appropriate license).

I would think taxes and perhaps duty would be owing to clear the phones in Albania. From this document, it looks to me (based on no ability to read Albanian) that the duty rate is 0% so just VAT at 20%, I think, should be owing. The information should be here, if you can read Albanian, and the information would cover any exemptions for visitors or for gifts.