Yes. If by that you mean can someone who is capable of calling the consumer from 7778588888 would be able to answer 777777777783 when it rings ... certainly.
But I think you should understand that the "outbound callerid" and the "inbound DID" are not related to each other any more (certainly not when you are using a VOIP telephone system).
INBOUND: You purchase a DID from your VOIP provider and when a call arrives at that DID, it will be routed to your Vicidial system and then to any agent you want.
OUTBOUND: The outbound callerid is entirely up to you. Obviously it would be good to use one that (when called) will result in a call to a DID you own. But you CAN put any callerid in the Campaign CallerID field and in the Admin->Phones=>{choose phone} "CallerID" field. Those entries are not regulated or monitored by most VOIP Providers. So .. you put in there anything you like.
But when the client calls that number it becomes an INBOUND call, and the owner of that number will get the call. Probably best to be sure YOU are the owner for any number you put in the CallerID field in your PBX.
Also, because I suspect you're new here and this may help:
1) Welcome to the Party!
2) As you are obviously new here, I have some suggestions to help us all help you:
When you post, please post your entire configuration including (but not limited to) your installation method and vicidial version with build.
This IS a requirement for posting along with reading the stickies (at the top of each forum) and the manager's manual (available on EFLO.net, both free and paid versions)
You should also post: Asterisk version, telephony hardware (model number is helpful here), cluster information if you have one, and whether any other software is installed in the box. If your installation method is "from scratch" you must post your operating system and should also post the .iso version from which you installed your original operating system. If your installation is "Hosted" list the site name of the host.
If this is a "Cloud" or "Virtual" server, please note the technology involved along with the version of that techology (ie: VMware Server Version 2.0.2). If it is not, merely stating the Motherboard model # and CPU would be helpful.
Similar to This:
Vicibox X.X from .iso | Vicidial X.X.X-XXX Build XXXXXX-XXXX | Asterisk X.X.X | Single Server | No Digium/Sangoma Hardware | No Extra Software After Installation | Intel DG35EC | Core2Quad Q6600
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Happy Hunting!