Are Tutti Bambini Cozee Basinets safe or do you need a bassinet with all mesh sides? I’ve been told by the company that distributes them, they are made with breathable fabric.

Red Nose recommends that the safest place to sleep baby is in a cot (with mandatory AS/NZS standards) in the parents’ room.

There are no standards for bassinets in Australia, including 3 sided which attaches to parents’ bed (side car cots) or the free-standing bassinets that have a side that can be opened.

Product Safety does have some recommendations (Buying Tips) for bassinets.
Check to see if the product you describe meets these safety recommendations.

Ensure the sides of the bassinette are at least 300 mm higher than the top of the mattress base.
Ensure it has a wide stable base and a sturdy bottom so that it won’t tip over.
Ensure the mattress fits well and is firm, smooth and no more than 75 mm thick.
If the product has folding legs, make sure it cannot accidentally fold during use.
https://www.productsafety.gov.au/products/babies-kids/kids-furniture/bassinettes

Are there any potential hazards such as entrapment, strangulation, suffocation or fall risks associated with using this product?

If the base of the bassinet does not fit flush with adult bed, there may be concern/risk of entrapment.
If the side is left down when baby is unsupervised, there may be a risk of falls.
As with all bassinets, once baby is attempting to roll, they need to be moved to a cot.

If a parent is thinking about using products where there are no AS/NZS Standards, it is important to ascertain not only whether the product may be effective but also is it safe to use under all circumstances and where the baby may be unsupervised

It is important for parents when they are thinking of buying a product for their baby to be able to make the best informed decision based on evidence of product safety. This is crucial for preventing SUDI or other fatal sleeping accidents.

It is often assumed that all nursery products sold in retail stores must be safe, however many products have never been formally tested so it can be difficult to tell what is safe for your child.
https://rednose.org.au/article/baby-safe-sleeping-products
https://rednose.org.au/article/bassinettes