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It obviously depends on firm and exact strategy but typically in prop trading capital is not a huge concern as most trading/market making in liquid products is high sharpe/low capacity. Some exceptions would be hedged spread positions (ETF or future versus cash) which can be both high sharpe/low capacity and capital intensive or a few desks at some prop traders may do longer horizon trades although I think it would be rare at most firms for a fresh grad to be responsible for that sort of longer term directional trade.
A trading desk gets capital, you don’t. Most of the time you will join an established desk instead of starting your own, which means you get whatever your desk can use.
I agree that this is how most firms look at things when measuring capital usage but I also doubt that the desk head is going to be happy if a junior trading is using a significant fraction of the desk's capital. I would expect that desk head may manage or provide guidance on capital usage for the different trades or team members on the desk to ensure the desk stays within the expected capital usage.
In the firm I am with (one of the two mentioned), desk members trade as one unified desk. It’s like we run a combined book together instead of each member having their own book.
Typically some form of leverage is being used, so the capital requirements are satisfied at the firm level, and at the desk or trader level they function as a sort of risk limit, among other risk parameters that may be limited.
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