Exact details remain to be established. The spectre of Austrian police officers (for instance) being able to spy on UK health or biometric records seems pretty remote - let alone the idea that, say, the Turkish government would have their own copies of UK databases as the Tories suggest. On the other hand, it isn't hard to imagine even a quite innocuous implementation of STORK protocols creating some interesting records.
Suppose you did go to France and tax a car using your UK.gov login - forget about ID cards. The French system would need to check with the Brit one that your eID was real, and a record of that check would exist in the relevant logs if nowhere else. A British official wishing to know what you were up to could easily search these files, and say "aha - chummy has just taxed a car in France", or been to see a doctor in Iceland**, or paid property taxes in Slovakia.
Likewise, our own Home Office might soon - if research next year determines that Stork-type ideas are feasible - have a somewhat better handle on just how many Estonians, Slovakians and Spaniards (etc) are using British government services and/or paying British taxes: and whether or not these people have a) obtained their mandatory foreign-resident ID cards and b) been truthful when submitting their details.
Lots of food for thought here. And it has to be said, those worried about the potential for catastrophic errors, bugs, leaks or backdoors in the government IT networks of the future may not find the compliant, ID-card-loving Belgians' choice of tech partner reassuring. ®
Update: Since publication the Home Office have been back in touch. They'd like to reassure everyone that there are "no plans" to share usage information between governments (except as necessary to authenticate eIDs). They also say that the UK Government Gateway doesn't keep any records of transactions "other than temporary ones" (though somebody in the government does - eg the tax people or the DVLA). Finally they say, in answer to whether you could use STORK eID checks to track an individual's activities is "categorically no."
So that's all right then.
*Breaks every rule of Acronym Club, we say. Nobody the Reg spoke to could say who came up with it. Presumably something like Secret Tracking And Law-Enforcement Kommissariat (STALK) didn't seem cuddly enough.
**Iceland is on board with Stork, though not an EU nation.