Been a while since I've truely been through the story so these maybe super inaccurate. May come back to this thread someday after going through the story again. Putting Shiki start and Shiki end to prevent Week 1 spoilers. Whether you consider one of them 'Shiki' or not is up to you.
Neku- INTP, atleast at the beginning. Very to himself, quick to adapt, and critical of everything around him. Has an easy time focusing on only what he is interested in.
Shiki (start)- ESFP Cute, bubbly, positive, loving, and friendly. Wanting to work with everyone and wants to help anyone who needs it.
Shiki (end)- ISFP Shy, sensitive, quiet, but would probably still wish to help people who need her.
Joshua- ISTP. Since it's hard to know who he was before the 1st week, we can only assume his personality before then and what could have motivated him to do what he did. As such this guess is from what we got to know of him from the 2nd week.
Beat- ESTP Easy going, not much to think of the future and lives only in the now. First to take action and acts before thinking
Rhyme- ENFP. Leaning more towards 'E' but could be 'I'. Could also be a 'T' instead of 'F'. First person to notice a change or danger and the first to act in response (for better or worse...) Very supportive and wants to help people be the best they can be.
Sanae- ENTP Quick to see a problem, quick to find the answer, and quick to put the answer into motion. Also easy for him to see most if not all the consequences of a cause.
Uzuki- ESTJ nuff said
Koki- INFP perhaps? I definitely see him as an 'I' rather than an 'E'. He also seems like he goes too much with the flow unless something ticks him the wrong way. (E.g. promises being undelivered *cough* Week 3 Day 3 or 4 *cough* *cough*) Which then he'll try and make it right.
Sho- INTJ Perhaps? I see him more as 'I' since he prefers working independently. Could see him 'P' rather than 'J'. (I believe at some point the game does say he and Neku are similar) Doesn't follow order, values, or traditions. When has something he's focused on, cares about nothing else and he goes all out on it. (I bet the reaper wished he put that effort into being the game master properly).