How to become compliant when selling cross-border to EU through marketplace(s)?

Although marketplaces in general bear the VAT liability, you should be aware of the regulation affecting you and how to handle it. 

In general, marketplaces are responsible for VAT liability when you are selling to the EU or UK through them. Especially important it is for non-EU or non-UK companies that are using the Low-Value Consignment schemes IOSS and UK VAT. In other words, they have to provide you with the IOSS / UK VAT number to use with applicable shipments. 

They are not, however liable for VAT on shipments over the thresholds. You have to set your shipping solution to support DDP  shipments where you collect all fees and taxes at checkout which the courier will then invoice you for. More expensive than DDU, but customer experience is better and you have significantly fewer returns.

If you are an EU company shipping intra-EU, you are liable for the VATs also when selling cross-border even if selling through a marketplace. For non-EU companies selling intra-EU via marketplaces, the VAT liability on the other hand lies with the marketplace.