They indeed pay no net VAT (it's not a cost for them in the sense of their profit and loss statement), but they do remit a bit of the VAT collected by the end consumer to their _local_ tax authority.
As an example, let's consider a VAT rate of 20%, and a Dutch company that buys from a French one and sells to a German one. Their costs per product are €80, and thus they pay €16 of VAT over that to their French suppliers. If they sell a product for €100 (i.e. they add €20 of value), then they collect €20 of VAT from their German buyers (which might in turn get it from the end consumers). There's a difference of €4 between what they received and paid in VAT, and that difference is collected by the Dutch tax authority. That €4 is not coincidentally the 20% VAT over the value added by the Dutch company.