In general, Trope Launch Pad items that are still getting regular attention from the person who initially proposed them ("original poster" or "OP") should not be significantly edited or launched by someone else without the OP's permission. This is because, while Trope Launch Pad items (like ordinary wiki articles) are technically free for any registered user to edit, they're also somewhat like a forum/discussion system, where you shouldn't edit someone else's comments at all.
The Up For Grabs tag indicates that OP doesn't care if someone else takes sponsorship of their draft.
It is also a common faux-pas to start a new Trope Launch Pad entry with the Up for Grabs tag. If you're in a rush to get your new trope out to launch, chances are you're overlooking something important in the process (maybe it Needs a Better Name, or Yes, We Do Have This One), which is an almost guaranteed way to get your trope pulled over in the Trope Repair Shop for reworking. You'll save the Repair Shop some trouble if you focus on writing a solid trope definition first; it's better to get a trope launched in good condition (no matter how long that takes) than it is to get it launched quickly.
Be aware that "Up For Grabs" does not mean "free to launch"—launching a draft, even one without a sponsor, without explicitly taking sponsorship yourself and announcing intent is called "rogue launching". Rogue launching is super rude, and it will get you suspended immediately.
When you take sponsorship of a draft, you need to rekindle discussion and resolve any lingering issues the previous sponsor didn't take care of—especially if the draft has been abandoned for significantly longer than the standard two months. While sponsors do occasionally abandon perfectly launchable drafts, it is far more common for an abandoned draft to need substantial work before it's ready.