It should be printed on the toothpaste itself at the very bottom normally raised up letters and numbers. I searched a bit and found that the ADA requires that toothpastes that contain fluoride to have an expiration date listed.
This is some information from Colgate:
"A shelf-life is determined only by the effectiveness of the ingredients in the toothpaste, so there's no direct danger to your health in using a recently expired tube in a pinch. But, it does impact the effectiveness of preventing tooth decay and cavities if you use expired toothpaste over the long term. Fluoride in toothpaste can start to break down, decreasing your protection from bacteria-causing decay. Colgate products typically have a two-year expiration date to ensure that the fluoride is at an optimal level of stability, as the ions in the fluoride are at their most effective within two years. After that time, the consistency in color and taste can change. Most pastes have a similar timeline, although some may have a shorter shelf-life. If you reach for a forgotten tube in your medicine cabinet, it's important to check the expiration date before using it."
Can't say that's the same for other toothpaste manufacturers but that's at least a bit of information.