You only need be concerned about your own uptime. If your counterparty is offline for extended periods, they are at risk of being cheated by you, but it does not put you at risk.
However, since forwarding payments on the shared channel requires both parties to be online, their being offline incurs an opportunity cost onto you due to your balance in that channel lying fallow without generating fees. As @sr_gi notes below, a peer that is frequently offline may also hamper your ability to use the Lightning Network at all, if they are one of your main connections to the wider network.