Stream of Letters to Words

Jelly, 28 bytes

“¤Œæ׿®¬©¥¤‘Jẋ"$FẋLẊ‘1¦+\Ṭœṗ

A monadic link taking a list and returning a list of lists.

Try it online! (the footer separates the resulting list of lists with spaces)

How?

Uses all the percentages in the distribution rounded to their nearest integer (thus being within the 0.69% allowed thresholds).

“¤Œæ׿®¬©¥¤‘Jẋ"$FẋLẊ‘1¦+\Ṭœṗ - Link: list (of characters), s
“¤Œæ׿®¬©¥¤‘                 - code page indexes = [3,19,22,17,11,8,7,6,4,3]
               $             - last two links as a monad:
            J                -   range of length = [1, 2, 3, 4, 5,6,7,8,9,10]
              "              -   zip with:
             ẋ               -     repeat list = [[1,1,1],...,[9,9,9,9],[10,10,10]]
                F            - flatten (into one list of length 100)
                  L          - length of s
                 ẋ           - repeat list (one list of length 100*length(s) with the
                             -              correct distribution of integer lengths)
                   Ẋ         - shuffle
                      ¦      - sparse application of:
                    ‘        -   increment
                     1       -   to indexes: 1 (offset indexes for the partition below)
                        \    - cumulative reduce by:
                       +     -   addition (e.g. [4,4,7,1,...] -> [4,8,15,16,...])
                         Ṭ   - untruth (yield a list with 1s at those indexes (1 indexed)
                          œṗ - partition s at truthy indexes (note: excess ignored)

PHP, 94 bytes

for(;$c=$argn[$k++];print$c." "[--$e])if(!$e)for($r=rand(0,71);$r>ord(x^"ywgSKAF:=?"[$e++]););

Run as pipe with -nR or try it online.

breakdown

for(;$c=$argn[$i++];            # loop $c through string
    print$c                         # 2. print current character,
        ." "                        # 4. if remaining length is 0, print space
        [--$e]                      # 3. decrement remaining length
    )
    if(!$e)                         # 1. if remaining length is 0,
        for($r=rand(0,71);              # get random value from 0 to 71
            $r>ord(x^"ywgSKAF:=?"[$e++])    # and increment $e while $r is > probability
        ;);

Note: ywgSKAF:=? represents the increasing probabities -1: [1,15,31,43,51,57,62,66,69,71]

Octave, 108 bytes

@(s)mat2cell(s,1,[k=(w=repelems(1:10,[1:10;'AMOKGEDCBA'-63])(randi(72,1,n=nnz(s))))(cumsum(w)<=n) n-sum(k)])

Try it online!

*Takes the string as input and outputs an array of strings.

*The last element of the output may be an empty string.