***Is It Better to Speak Softly or Loudly?

Neither is correct. The voice that is too loud hurts the listeners' ears -- the voice that is too soft is not being heard. In both cases, the message is being lost because listeners are unable to focus and it all boils down to volume.
I just received an email from a Public Speaking Forum in which the writer was questioning what to do with her client's loud voice. The responses from those who are professional speakers, as well as some who teach presentation skills, were interesting and varied -- none of which I would recommend.
It is important to realize that your volume level is a learned habit. Either your parents spoke softly or loudly and you imitated them or perhaps you did the opposite, hoping not to sound like a loud, boisterous mother or maybe a dad who was difficult to hear. Regardless of the reason, your inner ear is quite comfortable with your current volume.
The answer is two-fold:

1. You must retrain your inner ear to accept either more volume or less.

2. You must allow your chest to power your sound.
Once your chest becomes your primary sounding board, your volume will automatically decrease if you currently speak too loudly because sound vibrating in the mid-torso or chest cavity is warmer, richer, resonant. Yes, a loud voice being propelled by your throat, voice box, mouth, and/or nasal cavities tends to be strident, harsh, and has an edge. The resonant voice from the chest has no edge, no stridency, no harshness. It is warm like that of James Earl Jones, Barry White (Oh, Yes!), Sean Connery, Julia Ormond, Diane Sawyer, and Ashley Judd.
If you are soft-spoken, on the other hand, your volume will automatically increase, again, once your chest is powering your voice.
I am ever amazed when I see (and hear) one technique that can take two opposites and bring about the same results.
Loud Voice + Proper Voice Placement = Normal Volumer
Soft Voice + Proper Voice Placement = Normal Volume

It is also important to recognize that your lack of volume or preponderance of it labels you. If you speak too loudly, you may be considered arrogant, authoritative, headstrong, pompous or even obnoxious. Whereas, those who speak too softly are often looked upon as being weak, wimpy, immature, ineffective, or diffident. None of these adjectives may describe you fairly; but, that is the message you are sending.
Place you voice in your chest and just listen to the difference. You will sound better; you will probably look better; and, you most definitely will feel better about yourself.