Only execute twig block if a Matrix radio has a certain value
You can filter Matrix blocks on field values – in your case, the field "grouping":

{% for sponsor in sponsors.sponsors.grouping( 'Left column' ) %}


To conditonally output the whole wrapping div and section heading for the left and right aligned sponsors respectively, you can sort the sponsors into two arrays (left and right) beforehand, and then check the length property of those arrays before outputting the relevant markup.

Full example:

{# Pull all sponsors from the Matrix field #}
{% set sponsors = entry.sponsors.type( 'sponsors' ) %}

{# Filter sponsors into two arrays by "grouping" field value #}
{% set sponsorsLeft = sponsors.grouping( 'Left column' ) %}
{% set sponsorsRight = sponsors.grouping( 'Right column' ) %}

{# I can haz sponsors in the left column? #}
{% if sponsorsLeft | length %}
    &lt;div class='left'&gt;
    &lt;h2&gt;LEFT COLUMN TITLE&lt;/h2&gt;
    {% for sponsor in sponsorsLeft %}
        &lt;li&gt;
            &lt;a href="{{ sponsor.sponsorUrl }}"&gt;{{ sponsor.sponsorName }}&lt;/a&gt;
        &lt;/li&gt;
    {% endfor %}
    &lt;/div&gt;
{% endif %}
...
{# I can haz sponsors in the right column? #}
{% if sponsorsRight | length %}
    &lt;div class='right'&gt;
    &lt;h2&gt;RIGHT COLUMN TITLE&lt;/h2&gt;
    {% for sponsor in sponsorsRight %}
        &lt;li&gt;
            &lt;a href="{{ sponsor.sponsorUrl }}"&gt;{{ sponsor.sponsorName }}&lt;/a&gt;
        &lt;/li&gt;
    {% endfor %}
    &lt;/div&gt;
{% endif %}