I have reared cockroaches by the thousands, for years. Three species, but mainly the infamous Periplaneta americana which I am sure everyone has at least heard about (see picture at the end).
First of all: Cockroaches do bite, and they have powerful mandibles. This is in the paper below:
Many people think they won't bite because they have never handled cockroaches. In fact, cockroaches are amongst the most common insects scavenging dead bodies, and they'll pester sleeping individuals when they're starving (e.g. in crowded infested buildings or ships).
In the case of large P. americana, males are way more prone to biting than females, especially when they're flying around. I have been bit three times by these cockroaches, and it hurts. Once I was wearing latex gloves and seized a large male, and it bit me through the glove. Because of the pinch, I pulled it off with my other hand, and the glove teared open. Last time, I was bit about 1.5 year ago on the shoulder in my sleep by an adult female P. australasiae. The pinch was enough to wake me up (so I could kill the bloody roach) and left a red mark that lasted for 2 days.
There are numerous reports of cockroach bites going infected, specially around lips. Linked paper is referenced as: http://dx.doi.org/10.1590/S1413-86702006000100006
Finally, they're not poisonous neither venomous insects, as they produce no relevant toxins. However they may carry many pathogenic microbes, so laster effects from contact with cockroaches may develop.
Hope this was clear enough!