Do cancer cells give off specific chemical signatures?
The answer is it depends.  First, take a step back from the definitions of "cancer" versus "normal" cell and recognize that cells and tissues can undergo a variety of changes in their growth.  A solid cancer/tumor (as its usually defined) is something that grows and invades the surrounding tissue.   A spectrum of possible cell growth can be :

normal -> hypertrophy -> hyperplasia or dysplasia -> hamartoma -> benign tumor -> carcinoma in situ -> invasive cancer -> metastais

(this list is NOT suggesting these are the actual steps to cancer, but instead they are useful for the next paragraph): 

The point of this list is that from a human perspective we can see something odd happening when we see a tumor, but the view from inside a cancer cell is basically "things are fine, I'm just doing my cell thing".  

But, that said...when a tumor grows and cancer cells invade tissues a few things happen which can give rise to differences between cancer cells and normal tissues and which can be detected.
First, it needs energy, and cancer cells often switch to aerobic glycolysis (a phenomenon called the Warburg effect). As the link shows, a special form of labelled glucose can be used to detect this in the body.   

Secondly, a tumor invades the surrounding tissue and generates new blood vessels, and so enzymes that break down extracellular matrix  are released.   

Some tumors over-express and secrete markers (usually proteins) that can be detected.  The prostate specific antigen (PSA) and carcinoembryonic antigen are two examples.   

Other things secreted by certain tumors may include volatiles that can be detected by dogs.

The challenge with any 'signature' detection is the sensitivity and specificity  which often limits the use of tests that show some early promise.

Cancer cells DO have specific changes to their DNA that can be used as very specific signatures for detection.  Tumor DNA is shed into the blood and possibly the urine.

Finally, tumor biology overlaps with many aspects of normal biology, including wound healing and pregnancy and this continues to make it challenging to identify tumor specific markers that have high sensitivity and specificity.