This varies by jurisdiction. (Needless to say, this approach, even where permitted, is a bad choice for clients with thin pocketbooks.)
The American Rule
As a general rule, this is permitted in all U.S. jurisdictions without informing the attorneys about each other, and in general, there is no obligation of one of the attorneys who knows about the other to inform the other. Indeed, generally speaking, neither attorney is permitted to contact the other unless that is within the scope of the representation.
The exception to the general rule is that if one lawyer learns of the perpetration of a crime or fraud by the other, or of conduct in violation of the rules of professional conduct by the other, the lawyer learning of this fact may be allowed (in the case of a crime or fraud) or required (in the case of a violation of professional conduct rules) to report the misconduct to the appropriate authorities, at least if this can be done without harming the client, or if it is necessary to disentangle the lawyer contemplating making a report from being a participant in an ongoing crime or fraud.
The Continental European Rule
The general rule is different in most continental European jurisdictions. There it is a violation of the rules of professional ethics applicable to those lawyers to do legal work for a client who previously retained another lawyer and is not current on the client's bill with the previous lawyer, unless the matter is a time dependent emergency matter (e.g. a TRO necessary to maintain the status quo). Collateral to this ethical rule is the prudential consideration that lawyer B should contact lawyer A to confirm that lawyer A's bill in the matter is current.
Now, strictly speaking, the European rule does not forbid a client from simultaneously retaining two independent lawyers to render independent legal opinions on the same matter, and strictly speaking, the European lawyers are not generally required to discuss with each other if their respective invoices for legal services are current if there are other reasons not to doubt a client's assertion that this is the case, and certainly aren't required to discuss anything other than the status of their respective invoices unless the client authorizes communications about other matters (unless the lawyers suspect that this representation is being utilized to perpetrate a fraud or crime).
But, the European client probably is obligated to advise lawyers A and B that the other respective lawyer has been engaged so that they may evaluate their professional obligations regarding the payment of fees to prior lawyers accurately.