First, let's get crossposting from your own personal/company blog to Medium out of the way. Google frowns upon duplicate content. Posting content to your personal/company blog and then cross posting it to Medium may result in that post being penalized. Google wants to see one authoritative post that it can deliver in a SERP.
Second, when you publish to Medium, you lose control of your content. Medium now owns your content and Medium drives traffic to its own service - not yours. Sure, you can link to your site, but is anyone really going to click through after reading your Medium post? Probably not.
Third, when you're on Medium, you lose the frame of your company branding in your own website. Do you want readers going to Medium, or do you want readers to find your organization's voice on your own company blog? While Medium may look attractive and your company blog looks shabby, maybe it's time to think about why your company fails to maintain its online presence and needs to jump on someone else's solution, because they can't manage their own?
Fourth, you lose all the SEO value and page rank your site could have gotten, if you published your post there. You've now passed that on to Medium and they control the eyeballs.
Fifth, Medium is an aggregator. At the end of the day, you're going to need social media to push your content out to your readers. So, you're effectively sending people away from your site and to Medium to read something, instead of sending them to your site, where they could have learned more about what you do.
Now, that's not to say Medium or LinkedIn's publishing tools are not important for content marketers. If you are trying to get a thesis out into the marketplace, these services are effective tools to get your message published and disseminated. But, do not make a mistake in thinking that these services can replace your personal/company blog. They support your content marketing, but should never replace the real estate that you manage online.