Yes it's true. You’re referring to the various flagellant pilgrim bands that appeared sporadically in parts of Europe for the duration of the Black Death. Such penitent groups had been around since the mid-13th century and would continue into the 15th, though the plague seemed to increase their popularity. Not all territories warmly welcomed them, and ultimately Pope Clement VI forbade them, though this didn’t completely stop the practice. Most contemporary chroniclers of the plague mention them. In his eyewitness chronicle, the Benedictine abbot Gilles le Muisis gives one of the longer, more vivid accounts of one group’s arrival in Tournai. He claims they often traveled in groups of a hundred of more; the arrival of one group in a town often inspired citizens to form their own bands.

Here was the procedure when companies of two or three hundred, sometimes more, sometimes less, had gathered. It was their practice to wear over their clothes a sleeveless tunic called a cloche [i.e., bell-shaped cloak] in the vernacular. On the front of the tunic was a red cross on the breast and a similar cross on the back. It was cut on one side and hanging there were lashes–what we call scourges in the vernacular–that had three tips on them, each iron tip spiked with four needle-like points. They wore hoods on their heads and on top of them a cap on which was sewn a red cross front and back; they carried penitentiary staves in their hands. Entering the town they carried a cross, banners, and wax torches as they were able, some more, some fewer, and they went in order, singing in their native language–Flemings in Flemish, Brabantines in German, French in French. When they arrived before the image of Blessed Mary in the cathedral church they stopped singing. Afterwards they assembled in the town square or in the courtyard of St. Martin’s. They went into a private room or some other fitting place and stripped off their clothes and shoes, leaving someone behind to look after them until they returned. Then they came to the place with their heads covered by their hoods and the caps over them, their feet bare, and their bodies naked except for a cloth covering, like the kind bakers wear when they work. They wrapped this garment, which reached to the ground, completely around them over their breeches and cinched it at their waist. In their hands they held their scourges. . . . Then they started forming a circle and singing chants in their native language, which appointed singers began while the others responded in unison. Chanting and responding they made a circle and the chanters and responders whipped themselves. The hymn was such that they were required to prostrate themselves three times. When they reached the first verse, they all fell to the ground at the same time and, stretching their arms out at their sides, made the shape of a cross. Getting up on their knees, they variously tormented themselves with the scourges, so much so that those seeing them were astonished and wept and felt compassion for their suffering. They did this a second and a third time, the chanters (who were curates or religious mendicants of the various orders) singing in the middle of the circles that each led. After they had finished making these circles and punishing themselves with prostrations and beatings, after they had ended their chanting, they all knelt and one of the priests among them offered commendations and prayers, just as if he were in church at funeral rites or the end of a sermon. All of them stayed kneeling until he had finished his commendations and prayers. It was most pious and terrible for those present to see what prostrations and punishments they inflicted on themselves.

One of the surviving copies of his chronicle provides an illustration of what the bands looked like.

To answer your second question, the plague bacillus (Yersinia pestis) was not passed on from human to human by blood contact as far as I know. It was transmitted either by a flea bite or droplet infection (i.e., coughing or sneezing).

Sources: Gilles’ account from J. Shinners, Medieval Popular Religion: A Reader (2nd ed. 2007,p. 453-454). See Rosemary Horrox’s The Black Death (1994, pp. 150-54) for a long account of German groups and a short description of the flagellants in England and some analysis (96-97). Norman Cohn’s classic The Pursuit of the Millennium: Revolutionary Millenarians and Mystical Anarchists of the Middle Ages (1957) has an early, dated analysis.

EDIT: Added a missing verb