The first myth to settle with 1031 exchanges is that even though there certainly is an "exchange" of property in a 1031 exchange it is very unlikely that you will exchange property directly with either the purchaser of your existing property or the seller of the one you will eventually purchase.
The easiest way to think of a 1031 exchange is to think of the whole process as two simple transactions - one sale and one purchase. The purchaser of your rental property may buy it as a rental of their own or as a property to live in themselves. They may or may not be doing a 1031 exchange of their own. It makes no difference to you.
The seller of the property you eventually purchase may like-wise be selling you an investment property they own or perhaps the one they currently live in. In both cases it really boils down to these two parties filling out one additional piece of paperwork as part of the closing process. Neither the buyer of the property you are selling, nor the seller of the one you are buying, need concern themselves in any real way with any aspect of your 1031 exchange. For them it's a sales transaction like any other.
The "exchange" part happens on your end with what is called a "Qualified Intermediary" or "QI". The funds from the sale of your existing property sit with this entity until they can be exchanged into your new property. This is a simplified explanation but the concept is valid.
The rules basically state that you have 45 days from the time you sell your existing property to find your "exchange" property (or properties). You then have 180 days total to close on this new property (including the 45 days to find it - in other words you have 180 days total, not 45 + 180). When the replacement property is found the QI will assist you with the paperwork to make the two transactions a valid 1031 exchange.
Again, it is very important to note that the buyer of your old property and the seller of the one you are about to buy don't need to be involved in any meaningful way as part of the exchange. Too many investors think that their options are limited as to who they can sell to or buy from (or what kind of properties they can buy). As far as these parties to the transaction(s) are concerned it's like any other sale!
For some useful tips on 1031 exchanges see 1031 exchange tips.