You can use WebdriverJS to do that. Just create two Webdriver instances and navigate them, for example:
var WebdriverJS = require('webdriverjs'),
assert = require('assert'),
browser1 = new WebdriverJS({
desiredCapabilities: {browserName:'chrome'}
}).init().url('http://github.com'),
browser2 = new WebdriverJS({
desiredCapabilities: {browserName:'chrome'}
}).init().url('http://github.com');
browser1
.setValue('#js-command-bar-field',['webdriverjs','Enter'])
.getText('.sort-bar h3',function(err,text) {
assert(text.indexOf('found 24 repository results') >= 0);
})
.end();
browser2
.setValue('#js-command-bar-field',['linux','Enter'])
.getText('.sort-bar h3',function(err,text) {
assert(text.indexOf('We\'ve found 22,466 repository results') >= 0);
})
.end();
Two Chrome windows get opened and will execute your instructions independently.