There is no way to find exact value of signal s. you can estimate this signal if you knew  statistical behavior  of n1 and n2. for example, probability distribution of n1 and n2 or frequency bands of these noises.

In addition to what Hossein said, you can vectorize the problem, i.e.,

[x1 x2]T=[1 1]T*s+[n1 n2]T. Then maximum likelihood or least squares estimation may be used. 

Using a Kalman filter with a known model you can find 's' from a single equation. You can check this paper for reference

A. Routray, A. K. Pradhan, and K. P. Rao, “A Novel Kalman Filter for Frequency Estimation of Distorted Signals in Power Systems,” IEEE Trans. Instrum. Meas., vol. 51, no. 3, pp. 469–479, Jun. 2002.

This depends on how do you want to characterize the signal s; 

for each case there is an appropriate way; you can check :

- filtering operation

-source separation tools

- spectrum analysis methods and so;

See. Idea! You can get a rough decision. 

Exempl:  n1=5; n2=2; s=10;  x1=s+n1; x2=s+n2;

p=[1 1 0; 1 0 1; 2 1 1; 0 1 -1; 0 -1 1]  => inv(p'p) singular

p=[1 1 0.001; 1 0.001 1; 2 1 1; 0.001 1 -1; 0 -1 1]  => inv(p'p) nonsingular !!!  

C={p'*p); G=inv(C)

Y=[x1; x2; x1+x2; x1-x2; x2-x1]

B=p'*Y

A=G*B

A=[ 8.9989   5.9981  3.0011]    Bingo!!!   [10 5 2]