It ends up being like sales tax in the US. It is the store that pays the sales tax(note 1), but I have never seen a store just include the tax in the price, they always pass it on as an extra charge to the end customer. And this may not be a bad thing. It is probably good for the general public to be explicitly reminded how much tax they are paying.

1: You don't add up and pay the tax board every month right? In fact this is the central theme to successfully collecting taxes, never collect them directly from the public if possable. That is a hard thankless task. It is much easier to steal them from the much more easily policed companies, before the public sees the money in the form of income tax or when they buy in the form of a sales tax. As a specific example remember the "use" tax, you were supposed to do just that, add up and pay the sales tax for things you bought out of your sales tax jurisdiction, this proved impossible to collect so with the massive increase in sales out of the tax jurisdiction(cough, amazon, cough) the courts ordered that each company had to now keep track of and pay the sales tax for every infernal piddling little sales tax area, a huge hassle for them, but that's not the states problem and it is much easier to enforce than having each person do it.

It's generally not like a store, it's charged at time of import not at time of sale. The seller can choose to pre-pay for the importer (the customer) but they'll only do that if they know what to pay. If they don't know what to pay (i.e. if the tariffs change randomly with no notice) they'll tick the box that says importer pays and the buyer gets a call from customs when it arrives in the country saying pay the tariff or we impound/destroy your goods.