WARNING: Tube/Valve amplifiers use potentially LETHAL HIGH VOLTAGES.
Building, troubleshooting and testing of these amplifiers should only be
performed by someone who is thoroughly familiar with
the safety precautions around high voltages.
The first stages of an amp generally have the highest gain, hence, any noise or interference introduced into these stages will have the greatest impact on system performance. The shields shield against electrostatic fields and keep them from interfering with the electron flow inside the tubes.
When I designed my RIAA Phono amp I used skirted tube holders from the start due to the fact that the signal levels from the magnetic cartridge would be very low. I was glad I did because in use, if you lift any of the screening cans, hum and noise levels increase very noticeably. For well laid out amps with lower sensitivity and which require much higher input voltages, the need to screen is not so important.
Valves inside a screened metal case may not need individual screens. As well as removing electrostatic fields, the screen may have some effect on magnetic fields, and light. I read somewhere that photoemission can sometimes increase valve noise, although I guess it might depend on the extent to which the grids are visible.