You'll get a lot of opinions on what tire pressure to run but the correct tire pressure for you is not a matter of polling other rider's opinion. Here are the basics you'll need to decide for yourself.
Start with the BIKE (not the tire *see below) manufacturer's recommendation in the owners manual or under-seat sticker. This is the number they consider to be the best balance between handling grip and tire wear. Further if you're running alloy wheels on poor pavement, consider adding 2 psi to the recommended tire pressure just to reduce the likelihood of pothole damage. Just as you would for a car, increase the pressure 2 psi or so for sustained high speed operation (or 2-up riding) to reduce rolling friction and casing flexing. Check your tire pressure regularly, as they say.
In order to get optimum handling a tire has to get to its optimum temperature which is different for each brand of tire. Unless you own a tire pyrometer that will measure tire temperature directly, you’ll need to measure it indirectly by checking tire pressure since tire pressure increases with tire temperature. Tire temperature is important to know because too much flexing of the casing of an under-inflated tire for a given riding style and road will result in overheating resulting in less than optimum grip. Over-pressurizing a tire will reduce casing flexing and prevent the tire from getting up to the optimum operating temperature and performance again suffers. Sliding and spinning the tires also increase tire temperatures from friction heating.
A technique for those wanting to get the most out of their tires on the street is to use the 10/20% rule.
First check the tire pressure when the tire is cold. Then take a ride on your favorite twisty piece of road. Then, measure the tire pressure immediately after stopping. If the pressure has risen less than 10% on the front or 20% on the rear, the rider should remove air from the tire. So for example, starting at a front tire pressure of 32.5 psi should bring you up to 36 psi hot. Once you obtain this pressure increase for a given rider, bike, tire, road and road temperature combination, check the tire pressure again while cold and record it for future reference.
Each manufacturer is different. Each tire model is different. A tire design that runs cooler needs to run a lower pressure (2-3 psi front) to get up to optimum temperature. The rear tire runs hotter than the front tire, road and track. So the rear tire cold-to-hot increase is greater. Dropping air pressure has the additional side effect of scrubbing more rubber area.
When I used the tire pressures recommended by Ducati (32.5F/36R) for my 916 on my favorite road, I got exactly 10/20% on a set of Bridgestone BT-012SS. So I guess I'm an average rider and the BT-012SS runs at an average operating temperature compared to other brands.
For the track you'll have to drop the cold tire pressures an additional 10/20%. Track operation will get tires hotter (increasing the cold-to-hot pressure range) so starting at say 32/30 psi now should bring you up to the same temperature (and pressure) that 35/39 psi gave you for the street. Don't even think about running these low track cold pressures on the street.
Finally, dropping tire pressures on street tires for track use has its limitations, so street compound tires on the track often get too hot and go beyond sticky to greasy. That's why you have race tires. Race tire compounds are designed for severe operation at these higher temperatures for a limited number of thermal cycles. On the other hand, a race tire on the street usually won't get up to the appropriate temperature for good performance. At street speeds, the race compound often won't perform as well as a street tire.
Finally, a tire that is inflated to a lower pressure than recommended will have a tire profile that will sag slightly in the middle. This sagging profile results in increased rolling friction and causes the tires to run hotter. This will reduce tire life but it will also increase tire traction or grip. Depending upon racing conditions and the overall setup of the bike the increased grip may be necessary to be competitive even at the cost of tire life.
* Tire Manufacturer's Recommendations
Japanese sportbikes seem to have an extra 4-6 psi specified for their tires, compared to the equivalent Ducati. Why?
A tire manufacturer will recommend a pressure that is a balance between tire life and grip. When a bike manufacturer is developing a new model their test riders will determine what pressures in their opinion, best suit the new model. The recommended pressures are the best for general street (not track) riding, so you can increase grip somewhat by reducing pressures.
But to answer the question about higher recommended tire pressures for Japanese in-line fours versus Ducati twins - in-line fours heat up their tires more than a twin so a higher starting pressure is needed to prevent overheating the tires, particularly the rear tire.
Years ago, superbike racers discovered that it was easier to modulate the power to prevent wheelspin on the Ducati V-twins than it was to do the same on the Japanese inline-fours. This is because there is a longer interval (in terms of both time and crankshaft rotation) between cylinders firing, which gives the rear tire a break - time to recover traction and match its speed to that of the motorcycle.
More recently, more sophisticated traction control systems have been tried to reduce tire temperatures, improve tire life and lap times,,,