The steps I ended up using were as follows, and the comments and answers so far got me started in the right direction.

First I added a row to the "sitemap" table. Since we have multi-store set up, and because I want to keep my module store agnostic, I didn't hard code this INSERT into a MySQL migration, but just ran it on the store manually:

INSERT INTO sitemap (sitemap_type, sitemap_filename, sitemap_path, store_id)
    VALUES ('people', 'people.xml', '/sitemap/', 2);

Then I overwrote the Mage_Sitemap_Model_Sitemap model inside the global/models section in my own module's config.xml file:

<global>
    <models>
        <sitemap>
            <rewrite>
                <sitemap>Mymod_People_Model_Sitemap</sitemap>
            </rewrite>
        </sitemap>
    </models>
</global>

This overwrites any calls to Mage_Sitemap_Model_Sitemap site-wide with my custom model, but I didn't want to copy and paste too much code there. Using Petar Dzhambazov's suggestion, I used a conditional to defer to the parent class unless the sitemap_type is "people".

class Mymod_People_Model_Sitemap extends Mage_Sitemap_Model_Sitemap
{
    const PAGE_REFRESH_FREQUENCY = 'weekly';
    const PAGE_PRIORITY = '1.0';

    public function generateXml()
    {
        if ($this->getSitemapType() != 'people') {
            return parent::generateXml();
        }

        $io = new Varien_Io_File();
        $io->setAllowCreateFolders(true);
        $io->open(array('path' => $this->getPath()));

        if ($io->fileExists($this->getSitemapFilename()) && !$io->isWriteable($this->getSitemapFilename())) {
            Mage::throwException(Mage::helper('sitemap')->__('File "%s" cannot be saved. Please, make sure the directory "%s" is writeable by web server.', $this->getSitemapFilename(), $this->getPath()));
        }

        $io->streamOpen($this->getSitemapFilename());

        $io->streamWrite('<?xml version="1.0" encoding="UTF-8"?>' . "\n");
        $io->streamWrite('<urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">');

        $storeId = $this->getStoreId();
        $date    = Mage::getSingleton('core/date')->gmtDate('Y-m-d');
        $baseUrl = Mage::app()->getStore($storeId)->getBaseUrl(Mage_Core_Model_Store::URL_TYPE_LINK);

        /**
         * Generate people sitemap
         */
        $changefreq = Mymod_People_Model_Sitemap::PAGE_REFRESH_FREQUENCY;
        $priority   = Mymod_People_Model_Sitemap::PAGE_PRIORITY;
        $collection = Mage::getModel('people/person')->getCollection();
        foreach ($collection as $item) {
            $xml = sprintf('<url><loc>%s</loc><lastmod>%s</lastmod><changefreq>%s</changefreq><priority>%.1f</priority></url>',
                htmlspecialchars($item->getUrl()),
                $date,
                $changefreq,
                $priority
            );
            $io->streamWrite($xml);
        }
        unset($collection);

        $io->streamWrite('</urlset>');
        $io->streamClose();

        $this->setSitemapTime(Mage::getSingleton('core/date')->gmtDate('Y-m-d H:i:s'));
        $this->save();

        return $this;
    }
}

Is there a better way, that avoids copying and pasting so much from the parent class?

I don't know the page you want the Products menu item to point, but let's say that the page url identifier is products.
You need to change the line under the homepage link

<li><a href="<?php echo $this->getUrl('') ?>"><?php echo $this->__('') ?></a>

(I assume that in the line above <?php echo $this->__('') ?> should actually be <?php echo $this->__('Products') ?>).

to this:

<li><a href="<?php echo $this->getUrl('', array('_direct'=>'products')) ?>"><?php echo $this->__('Products') ?></a>