Are we allowed to start ones own currency complete with bills and coins?

Well, video-games , especially mobile games, are known for having virtual currencies that work only in specific games, to be bought for real money.
So, maybe?

Yes, I think as long as it doesn’t look like any country’s currency it would be okay. I use to go to a restaurant and bar that had wooden nickels for drinks on “Friday Happy Hour ”

Yes you are. I know people who have done this.

Yes, absolutely. Generically, it is called “scrip”.

The trick, however, is getting anyone to accept it. To be used, it has to have some inherent value. Something that you make would have no known value for exchange – no one would know what it was worth. So it wouldn’t be accepted.

Which, by the way, is why Bitcoin will ultimately fail. There is nothing that backs it up.

Private companies and people have made their own scrip that works as a medium of exchange. But it isn’t money, because it is not a store of value and it is not a unit of account. And it is usually limited in who will accept it.

Arcade coins and the like aren’t really currency any more so than poker chips are. Pretty much as Zenvelo said above.

Since Nixon took US currency off the gold standard, there is nothing backing up our money except the government saying they will cover it.

@elbanditoroso is right—scrip has been used in several social / societal experiments, usually to keep the money within the community. For instance, you can buy something at a farmer;s market or grocery or restaurant with scrip for less than what it costs with regular currency,

All answers here are excellentl Elbanditoroso makes several good points in his answer,

Take a look at the first paper money circulated in the States, when the government was fooling the world into thinking they knew what they were doing.
Our first paper money was completely worthless. An IOU carried more weight.
Legitimate transactions were made in British or French currency.

In current times one can print anything they want to and call it money, but an attempt to treat it like legal currency, such as using it to cover purchases or pay bills, or make a trade for actual legal tender would result in heavy disciplinary action.

@SEKA Nixon had to take us off the gold standard because there wasn’t enough gold to back up all of the currency needed to conduct the nation’s business.

@kritiper

Yep. All the gold mined in the history of the world would not cover the modern US economy.

@Darth_Algar @kritiper and yet we have distinguished politicians in 2019 who want the US to go back to the gold standard.

@elbanditoroso Yeah, but nobody ever proved that politicians have any smarts.