Whether or not you actually need to own a microphone depends on the kind of work you are doing. Those who play often may want to have more control over how they are heard, and owning your own microphone will give you that extra benefit, especially in live situations. In recording sessions however, it never hurts to use what the studio has, since there's a good chance that they've spent more money on quality microphones.
There are plenty of microphones that work superbly for studio recordings, but a common favorite amongst trumpet players is the ribbon microphone. Many well-known trumpet recordings used ribbon microphones, and the sound of a quality one is often very even and rich. However, you need to be careful with ribbon microphones, as they are quite fragile to handle and even a short burst of phantom power can destroy the ribbon altogether. Condenser microphones are also sometimes used for horns, but the sound can sometimes have too much high end and not enough body, especially when using mutes. For this reason, you're better off going with a ribbon microphone. A popular ribbon microphone brand is the R-122 made by Royer. For live performance settings, Royer also makes an R-121 version.
When playing live, there are a few things to think about. If you're the only trumpet player that's performing, you have more options. But if you're a part of a section, it's a good idea to make sure that everyone is playing on the same model. This way, you can be positive that the sound is matched perfectly across. In this situation, you might have to go with whatever the sound company has multiples of in stock. Very often, trumpet sections are mic'd with the classic Shure SM57. These mics not only work great, but they are affordable enough for sound companies to have numerous amounts in their arsenal. The Shure SM57 is also used commonly by solo players for club gigs; hence, you can't go wrong with purchasing your own.
If you perform as a soloist often, then having your own live microphone is ideal. These days, a popular choice of live microphones is the clip-on style. These microphones attach on to the bell of your horn, and a small goose-neck allows you to point the microphone directly into the bell. It even lets you play at any angle, which gives you more freedom to roam around the stage. Popular choices include the Audio Technica Pro-35, the AKG-C519M, and Shure Beta 98H.
It should be noted, however, that if you decide to go with a goose-neck clip-on, they can make quick use of mutes difficult, since they first must be moved out of the way, then repositioned. This could be even more troublesome for those who play in a Big Band. To resolve this issue, try going with the Applied Microphone Technology P800. Doing so will keep your bell free and clear for mutes. But keep in mind that any style clip-on will have the microphone attached to your bell at all times. So, if you like to perform at different distances from the microphone, you might want to avoid clip-on microphones altogether.
Remember, this article is to simply help you make a better decision. As mentioned above, your best bet is to try out as many different microphones as you can, and speak to other players and sound engineers. Doing so will ensure that your talents are being showcased to their highest degree of excellence; and in the end, that's what matters most.