Answer

A loan requires either witnesses (and according to some opinions, even one witness) or a document.
In any case, I am not aware of such a prohibition regarding a deposit.
Several differences between the cases can be suggested. For example, a borrower is usually needy, so there is more concern. Perhaps even with a loan one should distinguish between money and objects: money is meant to be spent, whereas objects are less fungible. Objects can be identified by their owner, while money cannot.
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Questioner:
Does the prohibition of lending without witnesses/a document also apply to very small amounts, like five or ten shekels?
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Rabbi:
In principle, yes, since the Sages did not differentiate in their enactment. Still, I should say that people are not generally careful about this, and usually the lender simply waives the debt if the borrower forgets and does not repay it. In that way, he is not causing him to stumble even if he does not repay. If you are not waiving it in any case, it is preferable not to do this.
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Questioner:
 
Is a monetary deposit similar to a loan, in the sense that it too must be given with a document or witnesses? For example, a deposit in a free-loan fund or a deposit in a bank. 
 
In addition, is there a prohibition against keeping at home a deposit document from a free-loan fund after they have returned part of the deposit to me? (“It is forbidden for a person to keep a paid document in his house,” Babylonian Talmud, Ketubot 19a.)
And what would the answer to the previous question be if only the recipient of the deposit signed the deposit document, and not two witnesses? Does that invalidate it?
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Rabbi:
I do not recall the halakhic decisors discussing deposits, even though there would seemingly be a similarity. But a deposit is not meant to be spent, so the likelihood of forgetting there is smaller. The question is whether the original enactment is concerned with intentional denial or with forgetfulness. (See Shakh and Sma at the beginning of sec. 70, where they discuss a Torah scholar, for whom there is no concern that he would intentionally deny it, but there is concern that he might forget.) If the deposit is spendable—that is, you do not need to return the exact money you received, only the same amount—then it is a loan and not a deposit.
But there is another difference between a deposit and a loan: someone who takes a loan is usually needy, whereas with a custodian, it is actually he who is doing a favor for the depositor. Perhaps with someone who is not needy, there is no concern that he will deny the deposit. This is true even if the deposit is spendable.
Bottom line: although I have not heard that people are careful about this—even with loans, people are lenient nowadays, though the halakhic authorities were puzzled by this—still, it is best to deposit with witnesses or some other proof. With a bank there is no problem, because there is evidence in the computer system and in their forms.
 
A document signed only by the obligating party is a valid document in every respect. (In the language of the Talmud and the halakhic authorities: his own handwriting.)
The prohibition against keeping a paid document in one’s house raises the same question I raised above: is the concern that he will forget, or that he will intentionally sue again? It seems to me that when an obligation is imposed on the person himself, it is more likely that the concern is that he may forget—for if he were acting intentionally, he would also violate the prohibition and keep it in his house. Therefore, to avoid forgetting, write on the document that part of it was paid, or have the free-loan fund give you a receipt for the amount that was paid.