I'm a little baffled by your question. I've made a simple simulation, data attached:

sum x1  x2  x3  x1_proportion   x2_proportion   x3_proportion   ones
1.44975 .884738 .331214 .233797 .6102698    .2284629    .1612673    1
1.75989 .793748 .655205 .310937 .4510212    .3722992    .1766796    1
1.35571 .462276 .882351 .011085 .3409837    .6508396    .0081768    1
1.63689 .002848 .708656 .925386 .0017398    .4329283    .565332     1
1.44575 .862857 .256457 .326439 .5968218    .1773864    .2257918    1
2.10639 .59055  .964992 .550847 .2803613    .4581261    .2615126    1
1.34527 .180885 .497332 .667048 .1344604    .3696907    .4958489    1
1.97426 .299043 .831939 .843283 .1514706    .4213918    .4271376    1
1.7669  .559657 .268161 .939079 .3167457    .1517697    .5314847    1
1.58345 .916163 .520577 .146706 .5785881    .3287623    .0926496    1
1.77596 .832321 .670544 .273091 .4686608    .3775677    .1537714    1
1.89561 .779795 .756137 .359681 .4113679    .398888     .1897441    1
.784696 .000545 .63612  .148031 .0006939    .810658     .1886481    1
1.63006 .25147  .58731  .791278 .1542705    .3603002    .4854293    1
1.8412  .526846 .327903 .986448 .2861431    .1780925    .5357644    1
1.52932 .627659 .802862 .098797 .4104179    .5249804    .0646017    1

Then in Stata (or whatever you want)

reg sum x1 x2 x3

obtains 1's for all $\beta$'s like anticipated and rounding error to 0 for the constant. $R^2$'s are all 1. But

reg ones x1_proportion x2_proportion x3_proportion, noconst

Obtains, again, all $\beta$s are 1. And if you allow the constant, you will get simply 1 for the constant and 0 for all $\beta$s. In either case, again, $R^2$'s are all 1.

I think you have some confusion about the meaning of regression here, so I will dig deeper and speculate a bit. I think you actually want:

... That x1 represents, on average 32.4% of the "sum" variable, with a standard deviation of 19.6%

I'm a little baffled by your question. I've made a simple simulation, data attached:

sum x1  x2  x3  x1_proportion   x2_proportion   x3_proportion   ones
1.44975 .884738 .331214 .233797 .6102698    .2284629    .1612673    1
1.75989 .793748 .655205 .310937 .4510212    .3722992    .1766796    1
1.35571 .462276 .882351 .011085 .3409837    .6508396    .0081768    1
1.63689 .002848 .708656 .925386 .0017398    .4329283    .565332     1
1.44575 .862857 .256457 .326439 .5968218    .1773864    .2257918    1
2.10639 .59055  .964992 .550847 .2803613    .4581261    .2615126    1
1.34527 .180885 .497332 .667048 .1344604    .3696907    .4958489    1
1.97426 .299043 .831939 .843283 .1514706    .4213918    .4271376    1
1.7669  .559657 .268161 .939079 .3167457    .1517697    .5314847    1
1.58345 .916163 .520577 .146706 .5785881    .3287623    .0926496    1
1.77596 .832321 .670544 .273091 .4686608    .3775677    .1537714    1
1.89561 .779795 .756137 .359681 .4113679    .398888     .1897441    1
.784696 .000545 .63612  .148031 .0006939    .810658     .1886481    1
1.63006 .25147  .58731  .791278 .1542705    .3603002    .4854293    1
1.8412  .526846 .327903 .986448 .2861431    .1780925    .5357644    1
1.52932 .627659 .802862 .098797 .4104179    .5249804    .0646017    1

Then in Stata (or whatever you want)

reg sum x1 x2 x3

obtains 1's for all $\beta$'s like anticipated and rounding error to 0 for the constant. $R^2$'s are all 1. But

reg ones x1_proportion x2_proportion x3_proportion, noconst

Obtains, again, all $\beta$s are 1. And if you allow the constant, you will get simply 1 for the constant and 0 for all $\beta$s. In either case, again, $R^2$'s are all 1.

I think you have some confusion about the meaning of regression (or I am not understanding the question) here, so I will dig deeper and speculate a bit. I think you actually want:

... That x1 represents, on average 32.4% of the "sum" variable, with a standard deviation of 19.6%