How can we ensure the correct lincensed manufacture of our product overseas?

I am patent attorney admitted to practice only in Japan. My advice below may not applicable to other jurisdictions so please ask advice to US attorney before you proceed. Please note that I will just explain general tips regarding this matter, and please understand I cannot bear responsibility to any consequences that may result by your conduct by following my explanation below.
Again, please talk to US attorney.

i) IP right. I focus on patent, not trademark.
I assume you have not acquired US patent or foreign patent covering your product. You can still file US patent application within 12 months from the time you started selling your product. So time is limited.
As to European patent, once you have started selling your product, there is no way you can patent your invention. (There is extremely limited exception, but that does not apply to your situation.)
So, making the long story short, if you want to secure your IP right, you should file US patent application within one month.
There is pitfall where Italian company start selling products before you file US patent application. Please refer to the following:
https://www.dlapiper.com/ja/japan/insights/publications/2013/03/the-aias-oneyear-grace-period--a-trap-for-the-un__/

Relating to my answer below, if Italian company wants to use you "know how," it is imperative to include in the license agreement the duty to keep the secrecy of the know how and the liability of the licensee when the know how is divulged to third party. Please see below:
http://www.iphandbook.org/handbook/ch11/p05/

Further, it may be useful to include in the contract the grant back provision, when Italian company patent some ideas or created other know how based on your know how. But be careful not to violate anti-trust law, by binding them to give you the exclusive right to use their newly developed know how. Please see below:
http://www.biodieselmagazine.com/articles/8888/the-grant-back-clause-in-your-technology-license

ii) It is unclear why Italian company wants to accept license if you do not have European patents.
I assume Italian company wants to use "know how" in your product. If so, you should be careful in the terms of contract with them.

"We are getting paid based on the correct number of manufactures items (how can we establish "trust")? "
You should include in the license contract duty for reporting and keep record;
http://entrepreneurship.org/resource-center/technology-licensing-ii-key-elements-of-an-agreement.aspx

Further, it is preferable to include your right to inspect their manufacturing record and accounting and audit, as explained in above web site.

Again, I disclaim responsibility for my explanation above. Please talk to US attorney before you make your decision.

The question is more business oriented than legal at my point of view.

You provide no information about if you have a patent or if even you can receive one for your products. So I will not develop this point without more information from you, and even a trade secret is acceptable both in US and in Europe.

So I guess that the problem is mainly about the trust and how to build through a contract this trust.

So there are plenty ways to touch this goal by adding specific contractual clauses with free access to accounting or companies who provide specific material to build the item.

The best way is still to keep a exclusivity in the manufacturing of one specific part of the item and to provide it to the European company. Then to request another European company to buy from time-to-time a commercial item manufactured to check if the specific piece of the item is yours.

Depending in the kind of item you build, we can maybe help you to realize this project.