Everybody reacts differently to bed bug bites.  Some people are more sensitive than others and some will develop an allergic reaction to the bed bug bites.  And then many people have no reaction at all to bed bug bites.
And to make things worse, many other insects will produce similar bite reactions such as insects, chiggers and spiders.
Although bed bug bite reactions will often appear as two or more bites in a row.
Often the bed bug bites look like a wheal i.e. a small swelling on the skin that usually itches or burns.
It can also form a blister and often results in a bacterial infection due to scratching of the bite.
And there typically is not a red spot at the center as with fleas of black flies.
There are lots of pictures of bed bug bites on the internet, but here are a couple to give you an idea.
A transition
Photo courtesy of Louis N. Sorkin, B.C.E., Entsult Associates, Inc
retromedial view forearm
Photo courtesy of Louis N. Sorkin, B.C.E., Entsult Associates, Inc
dorsal view of bite area
Photo courtesy of Louis N. Sorkin, B.C.E., Entsult Associates, Inc