Sightseeing is one of a class of English verbs that have two characteristics. First, they are intransitive because they already contain their object within the verb, and second, they always have an equivalent expression with the verb and object "unpacked". Other examples are: lovemaking, doorknocking, soothsaying, breakfasting, plus plenty of others. In each case, the verb doesn't take an object (lovemaking the bed would be bizarre), but it can be unpacked. So, one can say "they were breakfasting" or "they were breaking their fast". So, with sightseeing, the pattern is: they were sightseeing in the town, or they were seeing the sights in the town.
I don't think there's a problem in translation, if there is an exact equivalence of meaning. Rather that the English verb "sightsee" has to have "in the city" to be correct grammatically, rather than the exact Polish equivalent. If there's a lesson here, it's that sometimes between languages, it's impossible to get exact grammatical equivalence, but it is still possible to get exact equivalence of meaning. In this case if the Polish sentence describes someone going through the old town with a guide, and looking at the sights, then "sightseeing in the old town..." is the exact equivalent of the Polish.
Agreeing with some people above, I really don't think that "sightseeing the city" is a correct form ("in" is required). People from all sorts of language backgrounds speak English and so it may be emerging in some places as an acceptable form, but I don't think any English grammarian would say that it's correct yet.
The translation is grammatically incorrect in English. As mentioned above, 'sightsee' is an intransitive verb. It does not take an object. (My expertise: my degree is in English, and I get paid to write and edit in it.)
Please remove "sightseeing the city" as an acceptable translation. The default translation could be "sightseeing the city" or "seeing the city", both of which are grammatically correct.
I know that the translations of this sentence are far from perfect because it just doesn't translate very easily, but I have a problem with the idea of omitting "the city"... that is a really big change in the sentence - without it, maybe we are in a village, maybe we are just talking about one particular big building, like a cathedral?