The need for umbrella reflectors (usually around 5 to 7 inches in diameter) when used with umbrellas depends somewhat on the length of the flashtube and the design of the "face" of the monolight, in the area where the flash tube plug into the monolight itself. On some flash units, with long, large flashtubes (like Speedotron 100 series flash heads for example), when you use an umbrella without a 7 inch reflector, the flashtube disperses a pretty fair amount of light that goes all around the shooting area. On some monolights that have a very short, compact flash tube which is set into kind of a slight depression, that depression and the compact,short flash tube acts almost like a reflector. Some monolight systems have what is known as a "spill-kill" reflector, which is often a very short,compact,simple reflector that is designed to prevent spill light from flooding the shooting area.
Enclosed umbrellas like the excellent Lastolite Umbrella Box, or the Photek Softlighter, can be used with or without a reflector. I highly recommend the Lastolite Umbrella Box. it produces a nice light quality, and since the entire flash head is enclosed, it keeps all the light going "forward" toward the subject, and prevents a lot of stray light from bouncing around the shooting area. Lastolites are manufactured in the UK.
Silvered, metallized umbrellas, like the Speedotron Super Silver model for example, produce bright, crisp, somewhat specular light that will reflect a bit more off of skin on noses,foreheads, etc. This is an excellent type of umbrella for using on B&W shots. Dull-finished white interior reflecting umbrellas with black backs tend to give softer,more-diffused light that is less-specular than the Super Silver type umbrellas. Shoot-through umbrellas tend to send half of the light forward toward the subject, and half of ther light backwards, causing the spill light to go in all directions, especially in small rooms, or where the umbrella happens to be close to a wall; this can cause a LOT of what is called "ambient spill", which leads to a lower-contrast image and weaker color saturation and a sickly, cheezy light quality that I think looks sub-optimal. (I think very poorly of most shoot-through umbrellas, after having used them for quite some time.)
