In addition to blunt practice swords^(1), they did indeed also use sharp swords to practice. It was (and still is in some HEMA settings, especially in preparation for cutting competitions) necessary to do so, as a sharpened blade edge interacts differently with another surface than a blunt blade edge does.

One - and arguably the more obvious example - is that a sharpened blade edge allows for cutting. In order to cut properly the blade alignment (the angle at which the blade strikes the target) needs to be spot on - perpendicular to the target surface - otherwise the blade edge will be deviated away from the target surface along with all the kinetic energy, and this is something that needs to be practiced in order to master. While its use today is limited to cutting competitions and demonstrations (cutting tatami mats for example), there is little need for me to stress the importance of a proper cut in a martial medieval setting.

The second example of different handling characteristics has to do with how a sharpened blade edge has the tendency to bite into another surface, and how this facilitates binding: an interaction where friction between two blade edges is used to have one blade assert a degree of control over the other by obtaining superior leverage. This would be impossible to train with a blunt blade edge, as there is no virtually no biting, and therefore insufficient friction. Binding is prevalent in a large amount of treatises that handle sword-versus-sword combat, such as in I.33^(2), Fior di Battaglia^(3), the Codex Wallerstein^(4), and others.

^(On a side note: whereas not an academic source, if you would like to establish a clearer picture of what binding entails, how it depends on sharp blade edges, and why it necessitates practice in order to use in combat, I would recommend some of the videos on YouTube by Roland Warzecha, who demonstrates and practices with sharp edges.)

References

1. Joachim Meyers (1560s) - Some examples (1, 2, 3) from his Fäktbok , showing federn training swords.
2. Liutger (?) (1320s) - Some examples (1, 2, 3) from I.33, showing binding in sword-and-buckler fighting.
3. Fiore de'i Liberi (~1404) - Some examples (1, 2, 3) from Fior di Battaglia, showing binding in longsword fighting.
4. Unknown (relevant longsword section from the 1470s) - An example (1) from the Codex Wallerstein, showing binding in longsword fighting.