To give the short version to this answer, yes you need to check on several points. Lets first start with the obvious points. I have seen in my practice that reports that were given to a customer by a manufacturer where not always for the product that was being discussed. There are many reasons why this mixup is existing and the most positive are where the manufacturer is adapting a approved product to comply with his customers wishes. It happens on a regular basis that the manufacturer has made a new model based on and existing product and to make the sale he first uses the report for the existing model. Of course he will take care of testing the final model but during the proces the customer is already using the report that was made for the existing model and not the new one.
So first check if the product that you want to buy is really covered by the report that you have at hand. That is not always easy since you need to check details in the report. In general pictures are also included and these can often quickly reveal if this is the same product.
Second point to check is if the reports come from accredited laboratories. Everybody can put a sign above his door stating to be a laboratory. Does this mean that the company has the actual knowledge needed? No not necessarily. How do you check this? In the most countries in the world there is a accreditation organization. This organization will audit a laboratory on its capabilities, independence etc. If a laboratory complies with these requirements it gets an accreditation and that must be shown on there certificates. If you check the website of that accreditation organization it should be clear that the standard that is mentioned is actually listed. If this is the case you can be assured that the company is knowledgeable for this standard. It does not mean that they are able to test according to all standards you need so you should make sure that the accreditation is valid for all the standards that you need.
There are also other ways to establish if a company is actually knowledgeable. For example if you check the CB website you can see which labs have a CB accreditation for the standards that you need. There are two levels in CB. You can be a CBTL, which means that the company has testing capabilities and is able to test products. There are also NCB’s. This is a National Certification Body. This organization can issue a CB certificate but is not necessarily the organization that can perform the test. If a CB certificate is issued the NCB has to check if the lab for which they make the certificate is a CBTL.
On the level of European directives you also have labs that are being recognized. This is for the Low Voltage Directive a Notified Body. This is a organization that was audited and is knowledgeable in the application of the LVD. Certificates issued by these organization can be CE certificates for example. Please note that also CE certificate is mot protected and therefore anybody can make this kind of document. So also here is my advice, check check check.
In the end the importer in the EU is responsible for the compliance and safety of the products. Not the manufacturer outside of the union. So make sure that you do not get a problem due to misunderstandings or incorrect reports. You can not hide behind that!