The personality of the interpreter may be important in establishing trust with your business partners or employees and in providing a bit of sanity and perspective for you. The personalities of participants impact all interpersonal exchanges, and interpreters are participants in these exchanges. At the least, you want an interpreter whose personality will not antagonize the other party. Ideally you want someone whose personality harmonizes somewhat with your own. Two aspects of personality that are often particularly important are humor and warmth or empathy.
Appropriate humor is a very difficult thing to learn, far more difficult than language per se.
In many cultures humor is critical for building trust and thus it may be important to have appropriate humor in business transactions. For it to be useful rather than detrimental, the interpreter must have a keen sense of not only what is appropriate humor in the other culture, but what is appropriate in this particular situation. A good interpreter can warn you if your sense of humor is out of line or can break the tension by using his or her own humor. Warmth is something felt from the heart and transferred to others through verbal and non-verbal clues. The strangeness of the new situation can be mitigated through the warmth of a good interpreter.