Let's consider two trading partners, in this case Germany and the rest of the EU. (Among other things, having just two partners avoids the whole weighting issue.) The first line of the table below shows total demand in each -- that is, all private consumption, government consumption, and investment -- in billions of euros. (As usual, this is final demand -- transfers and intermediate goods are excluded.) So, for instance, in the year 2000 all spending by households, firms and governments in Germany totaled 2.04 trillion euros. The next two lines show the part of that expenditure that went to imports -- from the rest of the EU for Germany, from Germany for the rest of the EU, and from the rest of the world for both. The final two lines of each panel then show the share of total expenditure in each place that went to German and rest-of-EU goods respectively. The table looks at 2000 and 2009, a period of growing surpluses for Germany.
|Imports from EU||340||429|
|Imports from Rest of World||198||235|
|EU ex-Germany Share||17%||19%|
|Imports from Germany||387||501|
|Imports from Rest of World||795||998|
|EU ex-Germany Share||87%||87%|
|Ratio, Germany-EU Exports to Imports||1.14||1.17|
|EU Surplus, Percent of German GDP||2.27||3.02|
So what do we see? In 2000, 74 cents out of every euro spent in Germany went for German goods and services, and 17 cents for goods and services from the rest of the EU. Nine years later, 71 cents out of each German euro went to German stuff, and 19 cents to stuff from the rest of the EU. German households, businesses and government agencies were buying more from the rest of Europe, and less from their own country. Meanwhile, the rest of Europe was spending 4 cents out of every euro on goods and services from Germany -- exactly the same fraction in 2009 as in 2000.
If Germans were buying more from the rest of the EU, and non-German Europeans were buying the same amount from Germany, how could it be that the German trade surplus with the rest of Europe increased? And by nearly one percent of German GDP, a significant amount? The answer is that total expenditure was rising much faster in the rest of Europe -- by 2.7 percent a year, compared with 1.1 percent a year in Germany. This is what it means to say that the growing German surplus is entirely accounted for by demand, and that Germany actually lost competitiveness over this period.
Again, these are not estimates, they are the actual numbers as reported by EuroStat. It is simply a matter of historical fact that Germans spent more of their income on goods from the rest of the EU, and less on German goods, in 2009 than in 2000, and that the rest of the EU spent the same fraction of its income on German goods in the two years. Obviously, this does not rule out the possibility that German goods were becoming cheaper relative to the rest of Europe's, if you postulate some other factor that would have reduced Germany's exports without a growing cost advantage. (This is not so easy, since Germany's exports are the sort of high-end manufactures which usually have a high income elasticity, i.e. for which demand is expected to rise over time.) And it is also compatible with a story where German export prices fell, but export demand is price-inelastic, so that lower prices did nothing to raise export earnings. But it is absolutely not compatible with a simple story where the most important driver of German trade imbalances is changing relative prices. For that story to work, the main factor in Germany's growing surpluses would have to have been expenditure switching from other countries' goods to Germany's. And that didn't happen.
NOTE: This my table, not Enno's. The data is from Eurostat, while he uses the Penn World Tables, and he does not look at intra-European trade specifically.
UPDATE: There's another question, which no one asked but which you should always try to answer: Why does it matter? The truth is, a big reason I care about this is that I'm curious how capitalist economies work, and this stuff seems to shed some light on that, in terms of both the specific content and the methodology. But more specifically:
First, seeing trade flows as driven by income as well as price fits better with a vision of economy that has many different possible states of rest. It fits better with a vision of economies evolving in historical time, rather than gravitating toward an equilibrium which is both natural and optimal. In this particular case, there is no reason to suppose that the relative growth rates consistent with full employment in each country are also the relative growth rates consistent with balanced trade. A world in which trade flows respond mainly to relative prices is a world where macropolicy doesn't pose any fundamentally different challenges in an open economy than in a closed one. Whatever mechanisms operated to ensure full employment continue to do so, and then the exchange rate adjusts to keep trade flows balanced (or appropriately unbalanced, for a country with a good reason to export or import capital.) Whereas when the main relationship is between income and trade, they cannot vary independently.
Second, there are important implications for policy. Krugman keeps saying that Germany needs higher relative prices, i.e., higher inflation. Even leaving aside the political difficulties with such a program, it makes sense on its own terms only if there is a fixed pool of European demand. To say that the only way you can have an adequate level of demand in Greece is for prices to fall relative to Germany, is to accept, on a European or global level, the structural theory of unemployment that Krugman rejects so firmly (and rightly) for the US. By contrast if competitiveness didn't cause the problem, we shouldn't assume competitiveness is involved in the solution. The historical evidence suggests that more rapid income growth in Germany will be sufficient to move its current account back to balance. The implications for domestic demand in Germany are the opposite in this case as in the relative-prices case: Fixing the current account problem means more jobs and orders for German workers and firms, not higher inflation in Germany. 
So if you buy this story, you should be more pessimistic about a Greek exit from the euro -- since there's less reason to think that flexible exchange rates will lead to balanced trade -- but more optimistic about a solution within the euro.
I don't understand why, for economists like Krugman and Dean Baker, Keynesianism always seems to stop at the water's edge. Why does their analysis of international trade always implicitly  assume a world economy continually at full capacity, where a demand shortfall in one country or region implies excess demand somewhere else? They know perfectly well that the question of unemployment in one country cannot be reduced to the question of who is getting paid too much; why do they forget it as soon as exchange rates come into the picture? Perhaps it's for the same reasons -- whatever they are -- that so many economists who support all kinds of domestic regulation are ardent supporters of free trade, even though that's just laissez-faire at the global level. In the particular case of Krugman, I think part of the problem is that his own scholarly work is in trade. So when the conversation turns to trade he loses one of the biggest assets he brings to discussions of domestic policy -- a willingness to forget all the "progress" in economic theory over the past 30 or 40 years.
 A more reasonable version of the higher-prices-in-Germany claim is that Germany must be willing to accept higher inflation in order to raise demand. In some times and places this could certainly be true. But I don't think it is for Germany, given the evident slack in labor markets implied by stagnant wages. And in any case that's not what Krugman is saying -- for him, higher inflation is the solution, not an unfortunate side effect.
 Or sometimes explicitly -- e.g. this post has Germany sitting on a vertical aggregate supply curve.