<form>
<h1>Form 1</h1>
<div><input type="text" name="field1" placeholder="field1"></div>
<div><input type="text" name="field2" placeholder="field2"></div>
<div id="RecaptchaField1"></div>
<div><input type="submit"></div>
</form>
<form>
<h1>Form 2</h1>
<div><input type="text" name="field3" placeholder="field3"></div>
<div><input type="text" name="field4" placeholder="field4"></div>
<div id="RecaptchaField2"></div>
<div><input type="submit"></div>
</form>
In your javascript code, you have to define a callback function for recaptcha:
<script type="text/javascript">
var CaptchaCallback = function() {
grecaptcha.render('RecaptchaField1', {'sitekey' : '6Lc_your_site_key'});
grecaptcha.render('RecaptchaField2', {'sitekey' : '6Lc_your_site_key'});
};
</script>