Everything depends upon the infection and on the general immune status of the patient.
Generally, the prerequisite for DNA to freely circulate in the blood is the presence of bacteria themselves in the blood (bacteraemia). This means that the infection left its original site (where it is usually kept isolated from the blood flow by the immune system). Depending upon the body reaction to this breakthrough, sepsis and/or SIRS can be the consequences.
Under these conditions (not necessarily as severe as sepsis, but in case of proven bacteriemia), the bacteria cells get attacked by the immuno cells, that leads to their eventual lyzing and releasing their content to the blood.
PCR can be used as a method to prove the existence of bacterial DNA. (Here is a publicly available paper on this topic).