Does an electrocardiography recorder put current/voltage on its electrodes?
An ECG recorder works by recording electrical signals produced by the heart, and does not need to provide any active signal to do so.

That said, most commercial ECG systems will actually use a driven ground arrangement, called a "driven right leg" to make the recordings better (feeding back the common mode signal through a reference electrode), and thus do "send signals through the electrodes"  -- but those signals have nothing to do with the signals generated by the heart.  From an electrical engineering point of view, this minimizes the effect of different impedances between the different electrodes and the skin, among other things.

There's a reasonable review of the method at http://www.ti.com/lit/an/sbaa188/sbaa188.pdf

As pointed out by Zayzoon there may also be a small current for detection of leads falling off.

All these currents are not dangerous, and you can use the device without fear of injury.