you could test the melting point/boiling points. If they are spot on (well I guess oils you couldn’t do a melting point) then it’s pure. If they are contaminated, then the bp will be higher than listed.

toss the oil in the freezer or try distilling it.
The common carrier oils are grapeseed and sweet almond.
The freezing point of sweet almond is about 0f or -18c and grapeseed is close between -10 and -15c.
now expect to see some seperation of a pure essential oil since those oils are made up of several complex molecules. but if you have a diluted oil the usual ratio is 10% essential oil to 90% carrier, after a few days in the freezer you will end up with some seperation with a solid layer on the bottom and some liquid on top (assuming the essential oil has a lower freezing point). It takes a few days for the oils to seperate. this is an inaccurate test but is cheap and not terribley destructive to your oils.

Distillation is a much more accurate test and you don’t need much more than a thermometer.
Google a certificate of analysis for the oil you are buying. On the CofA it usually has the common components of the oil and their various boiling points. Make sure its the same plant source and expect some sometimes extreme variation in the component ratios, but the major boiling points should be accurate.
Use a thermometer to measuer the steam temperature as you apply heat to your sample and verify the various boling points. You can also get an idea of the ratios of the fractions by how long it boils off a particular product. If you run into anything unusual its probably a carrier oil. Since everything is dissolved together the boiling points wont be exact but the thermometer in the escaping steam will make sudden jumps in temperature as the fractions boil off.
Actual lab glass may be required and in the case of some essential oils the cost of the required sample size would make it cheaper to send a sample to a lab.

Edited to add

oops totally forgot the simplest one
place a drop of your supected oil on a peice of white paper and let sit for a day or 2
if it evaporates completely, or only leaves a little residue its pure, if it significantly loses smell but is still an oil spot its diluted.
This will only work for volatile oils, oils that are heavier than water will not usually evap at room temp properly. I’ve had that problem with styrax, dragonsblood and amyris sandalwood.

Labels are always good. When buying essential oils, the scent also comes as a factor in determining its grade. You might also consider asking around before buying essential oils cause some can become toxic if not handled properly.