I'm running the python code below which evaluates mean of an array:

def matlab_func1(array):
    p = os.popen('matlab -nodesktop -nosplash  -r "mean('+str(array)+');exit"')
    while 1:
        line = p.readline()
        if not line:
            break
        print line

matlab_func1([1,2,3])

From the matlab script below, it can be seen output return to y. I want to catch this output from python.

function y = mean(x,dim)
...
...
end

The solution must be applicable to the other matlab function. The 'mean' function is just an example.

Use fprintf to write needed text into stderr. Just add an extra argument 2 in the beginning.

import subprocess
import os
def matlab_func1(array):
    p = subprocess.Popen(['/home/user/Matlab/bin/matlab', '-nodesktop', '-nosplash', '-r "m = mean(' + str(array) + ');fprintf(2, \'%d\\n\',m);exit" >/dev/null'], stdout=subprocess.PIPE, stderr=subprocess.PIPE)
    while 1:
        try:
            out, err = p.communicate()
        except ValueError:
            break
        print 'hello' + err

matlab_func1('[1,2,3]')

A few things to note:

Changed the Python command to subprocess.Popen, which allows stderr piping.
In Matlab command, use fprintf to write wanted info into stderr. This can separate code ouput from Matlab's header lines.
Back into Python, use Popen.communicate() to catch stderr output.
The exception ValueError handles exit event of Matlab (p is closed).

EDIT:

for a function that gives multiple outputs

say a Matlab function is

function [y, z] = foo(x)
    y = x+1;
    z = x*20;
end

The point is use fprintf to throw the output, while doing all other things as you always do normally in Matlab.

Method 1 - in-line script

p = subprocess.Popen(['/home/user/Matlab/bin/matlab', '-nodesktop', '-nosplash', '-r "[y, z] = foo(' + str(array) + ');for ii=1:length(y) fprintf(2, \'%d %d\\n\',y(ii),z(ii)); end; exit" >/dev/null'], stdout=subprocess.PIPE, stderr=subprocess.PIPE)

Method 2 - standalone script

First create a new caller.m script

[y, z] = foo(x);
for ii=1:length(y)
    fprintf(2, '%d %d\n',y(ii),z(ii));
end

Note that x is to be assigned when calling from Python; scripts share the same stack. (Remember NOT to clear the workspace in the caller script. )

Then, call the script from Python

p = subprocess.Popen(['/home/user/Matlab/bin/matlab', '-nodesktop', '-nosplash', '-r "x=' + str(array) + ';caller; exit" >/dev/null'], stdout=subprocess.PIPE, stderr=subprocess.PIPE)

Storing Matlab result in a Python variable

When passing data through stdout / stderr pipe:

Refer to this and subprocess.check_output().
When handling serious data like double or binary:

Write the data into an external file with Matlab. Then read this file with Python. A protocol with which both side talk with each other should be defined.