Is it allowed to operate an aircraft when the Emergency Locator Transmitter has been removed?
Assuming that you are in the US, you need to comply with 

§91.207   Emergency locator transmitters. 

(2) … there must be attached to the airplane an approved personal type or an approved automatic type emergency locator transmitter that is in operable condition, 

and

(ii) No person may operate the aircraft more than 90 days after the ELT is initially removed from the aircraft; and

So in the next 90 days you need to install an ELT. It must meet the TSO standards for ELTs.

The cheapest way to comply would be to purchase a used 121.5 MHz unit so you are legal and use a personal unit in case of an accident. 

The newer 406MHz units require a GPS source and a panel switch. Installation costs can be more than the unit. However, since the previous owner removed the ELT, I would assume that the wiring is still intact so you can probably save money on the install.

Which type of attached transmitter you choose depends on your budget and the terrain over which you fly. On February 1, 2009, the international COSPAS-SARSAT satellite system discontinued satellite-based monitoring of the 121.5/243-MHz frequencies, in part because of a high number of false signals attributed with these frequencies. AOPA

They also frequently fail to activate and since they are only monitored by pilots who are following the AIM and monitoring 121.5 even if they do go off, they often are not heard. Not to mention the problem of triangulating the signal to find the source. If you fly over populated areas and are in contact with ATC via flight following or IFR flight plans, they are still adequate and most pilots I know have not upgraded to the 406 MHz units.

The newer 406 MHz are satellite monitored and because they transmit GPS location data it is easier to find a downed aircraft. They have an activation switch in the panel so that they can be activated before the pilot hits terrain which increases the odds of finding you. If I frequently flew over mountains, remote areas, or at night I would certainly install one of these.

Personal beacons have lots of advantages over installed ELTs. If you survive the landing, chances are that they will as well. They are also incredibly cheap and many of them have web-tracking or texting that can be used by those interested in tracking your progress. You can also take them with you if you leave the airplane.