A simple contactless method suitable for discerning between the intergrain ͑circular͒ current, which flows in the thin superconducting ring, and the intragrain current, which does not cross the weakest link, has been proposed. At first, we show that the intergrain current may directly be estimated from the magnetic flux density B(Ϯz 0 ) measured by the Hall sensor positioned in the special points Ϯz 0 above or below the ring center. The experimental and numerical techniques to determine the value z 0 are discussed. Being very promising for the characterization of a current flowing across the joints in welded YBaCuO rings ͑its dependences on the temperature and external magnetic field as well as the time dissipation͒, the approach has been applied to study the corresponding properties of the intragrain and intergrain currents flowing across the a-twisted grain boundaries which are frequent in bulk melt-textured ͑MT͒ YBaCuO samples. We present experimental data related to the flux penetration inside a bore of MT YBaCuO rings both in the nonmagnetized, virgin state, and during the field reversal. The shielding properties and their dependence on external magnetic fields are also studied. Besides, we consider flux creep effects and their influence on the current redistribution during a dwell.