I just toss 'em. Plastic or paper, disposed of appropriately with label intact. My name and address are publicly available anyway.

Parcelforce track and trace is a complete joke, no more than a certificate of posting

I use a black Sharpie Magnum to obliterate personal data on such labels. Sometimes it is still possible to see the underlying print if looking at the right angle; in that case, I further obliterate by scribbling with a biro.

Scoakat - in the UK, personal info isn't as widely available as in the US (and there is even the option to remove your details from the publicly available electoral roll). As many postage labels include name, address and telephone number, official advice (to avoid identity fraud) is not to put them into the bin without destroying/obliterating them in some way.

Also, if you naughtily put things into your rubbish bin that your rubbish collecting local authority might not approve of, the presence of identifying information could lead to A Visit From Officialadom, and potentially a large fine.

I put plastic bags into my general rubbish - ours goes to the incinerator so I'm hoping there's less opportunity for identity theft.
Our local Evri has improved enormously lately. I'm thinking it must be a new contractor but whatever it is long may it last.
Some Christmas presents I ordered from John Lewis were a doddle as I can get them delivered for free to the Waitrose at the end of my street.