You still use it on your own, so personal.

Anyway, once you abolish private property, most versions of 'working for someone else' go too, so this is somewhat academic.

But, the end product belongs to the person I was working for. It's still personal property?

>once you abolish private property, most versions of 'working for someone else' go too

How so?

Well if you are just making something for someone else, that's artisan labour, which isn't really the 'production' that's being analysed by Marx.

'Working for someone else' as found in 'employment' is also something that marxists seek to abolish. Collective ownership of the means of production also means collective ownership of the product of labour. You only have 'employment' per se when ownership of capital is concentrated in private hands.

(It's not for nothing that many marxists classify the former soviet union and post-Mao china as 'state capitalist' rather than 'socialist').